Hamiltonian formalism in Friedmann cosmology and its quantization
Jie Ren1,∗ Xin-He Meng2,3, and Liu Zhao2
Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China
Department of physics, Nankai University, Tianjin 300071, China and
BK21 Division of Advanced Research and Education in physics, Hanyang University, Seoul 133-791, Korea
(Dated: October 15, 2018)
We propose a Hamiltonian formalism for a generalized Friedmann-Roberson-Walker cosmology
model in the presence of both a variable equation of state (EOS) parameter w(a) and a variable
cosmological constant Λ(a), where a is the scale factor. This Hamiltonian system containing 1
degree of freedom and without constraint, gives Friedmann equations as the equation of motion,
which describes a mechanical system with a variable mass object moving in a potential field. After
an appropriate transformation of the scale factor, this system can be further simplified to an object
with constant mass moving in an effective potential field. In this framework, the Λ cold dark matter
model as the current standard model of cosmology corresponds to a harmonic oscillator. We further
generalize this formalism to take into account the bulk viscosity and other cases. The Hamiltonian
can be quantized straightforwardly, but this is different from the approach of the Wheeler-DeWitt
equation in quantum cosmology.
PACS numbers: 98.80.Jk,45.20.Jj,03.50.-z
I. INTRODUCTION
Since the current accelerating expansion of our Uni-
verse was discovered [1] around 1998 and 1999, theoret-
ical physicists have devoted increasingly more attention
to the Friedmann-Roberson-Walker (FRW) model as a
standard framework in cosmology study. The Λ cold dark
matter (ΛCDM) model as the standard model of cosmol-
ogy so far fits well with observational data whereas it
has had some serious theoretical problems. To make a
comparison to the ΛCDM model, physicists have built
many cosmological models that are able to give out the
effective Friedmann equations with variable cosmological
constant. To quantize the Friedmann equations, the com-
monly used theory is the Wheeler-DeWitt equation [2],
which has been studied and applied widely in quantum
cosmology [3]. Starting from the Hilbert-Einstein action
with the Roberson-Walker (RW) metric, the correspond-
ing HamiltonianH can be obtained. Then the Friedmann
equation plays the role as the constraint H = 0, which
leads to the Wheeler-DeWitt equation. In the present
work, we consider the Friedmann equations as basic equa-
tions and find a Hamiltonian system that gives Fried-
mann equations as classical equations of motion without
constraint.
Many Ansätze of the variable cosmological constant
have been studied in the literature, such as Refs. [4, 5,
6, 7]. Moreover, some models motivated from the string
theory give an effective cosmological term when reduced
to the FRW framework. We assume that the equation
of state (EOS) parameter w ≡ p/ρ can also be variable,
which means that the contents of the Universe, except
the cosmological term, are generalized to a nonperfect
∗Electronic address: jrenphysics@hotmail.com
fluid, or perfect fluid as a special case. In observational
cosmology, the redshift z is regarded as an observable
quantity and related to the scale factor a by z ≡ a0/a−1.
Therefore, we investigate a general case that both the
EOS parameter w and the cosmological constant Λ can
be functions of the scale factor a, and take into account
the bulk viscosity.
As an extension of the problem, we construct a Hamil-
tonian formalism for a system described by the following
equation:
q̈ = f1(q)q̇
2 + ηq̇ + f2(q),
where f1(q) and f2(q) are arbitrary functions, and η is
constant. Also it can be regarded as a generalization
of the damping harmonic oscillator. The corresponding
Hamiltonian describes an object with variable mass mov-
ing in a potential field. After an appropriate canonical
transformation, this system can be further simplified to
an object with constant mass moving in an effective po-
tential field. Thus, differential models in the FRW frame-
work are characterized by their effective potentials. This
is a general formalism and it can be applied to many cos-
mological models, for example, that the ΛCDM model
corresponds to a harmonic oscillator. Since the quanti-
zation of Friedmann equations can provide an insight to
quantum cosmology as a glimpse of quantum gravity, we
also make some remarks on the quantum case, which pro-
vides a correspondence between cosmology and quantum
mechanics.
The paper is organized as follows. In Sec. II we present
a generalized FRW model and the corresponding Hamil-
tonian to describe the Friedmann equations. Then we
find a canonical transformation to further simplify the
problem, and give some examples and special cases. In
Sec. III we show that our framework can also be applied
in the dissipative case with bulk viscosity. In Sec. IV
we turn our attention to the relation to the observable
http://arxiv.org/abs/0704.0672v3
mailto:jrenphysics@hotmail.com
quantities and review some issues of the Bianchi identity.
In Sec. V we make some remarks on quantum cosmology
from our approach. In the last section we present the
conclusion and discuss some future subjects.
II. HAMILTONIAN FORMALISM
A. Hamiltonian description of the Friedmann
equations
We consider the RW metric in the flat space geometry
(k=0) as the case favored by current cosmic observational
data:
ds2 = −dt2 + a(t)2(dr2 + r2dΩ2), (1)
where a(t) is the scale factor. The energy-momentum
tensor for the cosmic fluid can be written as
T̃µν = (ρ+ p)UµUν + (p+ ρΛ)gµν , (2)
where ρΛ = Λ/(8πG) is the energy density of the cos-
mological constant. Thus, Einstein’s equation Rµν −
gµνR = 8πGT̃µν contains two independent equations:
, (3a)
= −4πG
(ρ+ 3p) +
. (3b)
The EOS of the matter (cosmic fluid except the cosmo-
logical constant) is commonly assumed to be
p = (γ − 1)ρ. (4)
Cosmologists usually call Eq. (3a) as the Friedmann
equation and Eq. (3b) as the acceleration equation in the
literature, whereas for simplicity we name both Eqs. (3a)
and (3b) Friedmann equations here. For generality, we
assume that both γ and Λ are functions of the scale factor
a, thus we call it the generalized FRWmodel. Combining
the Friedmann equations with the EOS, we obtain
= −3γ(a)− 2
γ(a)Λ(a)
, (5)
which determines the evolution of the scale factor.
We regard Eq. (5) as a basic starting point; therefore, if
the dynamical equation for the scale factor can be written
as that form, the present framework can be valid. If
the Newton constant G is constant and the cosmological
constant Λ is variable, the energy-momentum tensor for
the matter cannot individually conserved [5, 6], which
implies an interaction between the matter and vacuum
energy. In the following, we assume G to be constant
until Sec. IV.
Our aim is to find a Hamiltonian description of Eq. (5)
as the classical equation of motion. We start from the
following Lagrangian
L(q, q̇) = 1
M(q)q̇2 − V (q), (6)
and the corresponding Hamiltonian thus is
H(q, p) = p
2M(q)
+ V (q), (7)
with the canonical Poisson bracket {q, p} = 1. One can
check that the equation of motion for Eq. (6) or (7) is
∂ lnM
∂ ln q
. (8)
This equation possesses the same form as Eq. (5). There-
fore, by comparing Eq. (5) with Eq. (8), we can take a as
the general coordinate and solve the functions M(a) and
V (a). Then the Lagrangian L = 1
M(a)q̇2 − V (a) with
M = exp
3γ − 2
, V = −1
MγΛada, (9)
gives Eq. (5) as the equation of motion. For some spec-
ified functions γ = γ(a) and Λ = Λ(a), the above in-
tegrations can be evaluated out to give M(a) and V (a)
explicitly.
Now we can see that the generalized FRWmodel essen-
tially corresponds to an object with variable mass M(a)
moving in a potential field V (a). In the following, we
will show that this picture can be further simplified as
an object with constant mass moving in an effective po-
tential field Ṽ (φ), after an appropriate transformation of
the scale factor.
B. Canonical transformation
The above problem can be generalized as the Hamilto-
nian description of the nonlinear equation
q̈ = f1(q)q̇
2 + f2(q), (10)
where f1(q) and f2(q) are two specified functions.
This equation can be derived by the Lagrangian L =
M(q)q̇2 − V (q) with
M = exp
f1(q)dq
, V = −
Mf2(q)dq. (11)
We define a new variable φ as (see Appendix)
f1(q)dq
dq. (12)
This transformation can eliminate the q̇2 term and gives
the equation for the variable φ as in
φ̈ = f2(q) exp
f1(q)dq
, (13)
where q → φ denotes using Eq. (12) to change the vari-
able q to φ. Since there is no φ̇2 term in Eq. (13), this can
be regarded as a partial linearization. Therefore, the sys-
tem of Eq. (10) transformed to Eq. (13) can be described
by the Lagrangian
L(φ, φ̇) = 1
φ̇2 − Ṽ (φ), (14)
with the potential as
Ṽ (φ) = −
f2(q) exp
f1(q)dq
f2(q) exp
f1(q)dq
.(15)
The simplification of the problem by Eq. (12) is essen-
tially the canonical transformation
q → φ, pq → pφ, H(q, pq) → H(φ, pφ), (16)
where pq = M(q)q̇, pφ = φ̇, and H(φ, pφ) = 12p
φ + Ṽ (φ).
Therefore, the classical and quantum properties of differ-
ent models are characterized by the effective potentials.
For Eq. (5) as a special case, the new variable φ is
given by
3γ − 2
da. (17)
C. Some examples
We will give some special cases of the above general
framework to show some applications. If both γ and Λ
are constant for a simple case, the integrations in Eq. (17)
can be evaluated out as
a3γ/2, γ 6= 0, (18a)
= ln a, γ = 0. (18b)
Now we consider γ 6= 0 for example. The special case γ =
1 corresponds to the ΛCDM model. The equation for φ
can be obtained as φ̈− 3
γ2Λφ = 0, and the corresponding
Lagrangian is
L = 1
φ̇2 +
γ2Λφ2. (19)
We can see that the simplest model in cosmology just
corresponds to a harmonic oscillator after linearization.
In particular, this is a upside-down harmonic oscillator
for the asymptotic de Sitter Universe.
We can add the curvature effect to the ΛCDM model,
which is described by the special case m = 2 of the fol-
lowing equation:
= −3γ − 2
. (20)
Here the parameters γ, Λ, and m are all constants. This
equation possesses the same form of Eq. (10). By defining
φ as Eq. (12) and using Eq. (15), we obtain the effective
potential as
Ṽ (φ) = −3
γ2Λφ2 +
3γ −m
)2−2m/3γ
, (21)
for γ 6= 0 and m 6= 3γ.
Another example is the Friedmann equations during
the inflation era. In the study of inflation, we usually
use the conformal time τ instead of the comoving cosmic
time t. Here we assume that a constant term −p0 is in
the EOS during inflation. Thus the Friedmann equations
combined with the EOS p = −ρ− p0 yield
p0, (22)
where the prime denotes a derivative with respect to τ ,
and κ2 = 8πG. By defining φ = −1/a, the equation for
φ is φ′′φ+ (κ2/2)p0 = 0. The effective potential is thus
Ṽ (φ) =
p0 ln |φ|. (23)
Moreover, if we add the curvature term in this case, it
corresponds to a φ2 potential.
III. BULK VISCOSITY
We assume that the cosmic fluid possesses some dis-
sipation effects. Since the sheer tensor σµν = 0 for RW
metric, the sheer viscosity does not contribute to the evo-
lution in Friedmann cosmology. The energy-momentum
tensor for nonperfect fluid concerning bulk viscosity in
the right-hand side of Einstein’s equation is given by [8, 9]
Tµν = ρUµUν + (p− ζ0θ)hµν , (24)
where hµν ≡ gµν + UµUν is the projection operator,
θ ≡ Uα;α = 3ȧ/a is the scalar expansion, and ζ is the
bulk viscosity coefficient. Consequently, Eq. (5) should
be modified as
= −3γ(a)− 2
+ 12πGζ0
γ(a)Λ(a)
. (25)
where both γ and Λ can be functions of a for generality,
and ζ0 is constant. We also find a Hamiltonian
H(a, pa, t) =
2M(a, t)
+ V (a, t), (26)
with the Poisson bracket {a, pa} = 1 to give Eq. (25) as
the classical equation of motion. The functions in this
Hamiltonian are given by
M = exp
3γ − 2
da− 12πGζ0t
, (27a)
V = −1
MγΛada. (27b)
Although a dissipative system cannot be described by
a conservative Hamiltonian generally, one can directly
check that the classical equation of motion for the Hamil-
tonian Eq. (26) is Eq. (25). As a special case, the equa-
tion for a damping harmonic oscillator can be derived by
the Caldirora-Kani (CK) Hamiltonian [10].
The above problem can be generalized to construct a
Hamiltonian system for the equation
q̈ = f1(q)q̇
2 + ηq̇ + f2(q), (28)
where η is constant. It can be derived by the Hamiltonian
H(q, p, t) = 1
M(q, t)−1p2 + V (q, t) with
M = exp
f1(q)dq − ηt
, V = −
Mf2(q)dq.
Similarly, by using the new variable φ defined by
Eq. (12), the equation for φ is
φ̈ = ηφ̇+ f2(q) exp
f1(q)dq
. (30)
Now we consider a very special case that both γ and Λ
are constant; then φ defined by Eq. (18a) satisfies
φ̈− 12πGζ0φ̇−
γ2Λφ = 0, (31)
which describes a damping harmonic oscillator.
The damping harmonic oscillator
Mq̈ = −ηq̇(t)−
∂V (q)
, (32)
has been studied in quantum mechanics. The CK Hamil-
tonian
e−ηt/Mp2 +
Mω2eηt/M q2, (33)
with the commutation relation [q, p] = i~, can yield the
dissipation equation (32) through the Heisenberg equa-
tion [10]. Our work can be regarded as a generalization
to the case of variable mass. It is the variable mass that
generates a nonlinear term in the equation of motion that
describes the generalized FRW model.
In our previous work [9], we have proposed an EOS as
p = (γ − 1)ρ−
κ2T 22
where the parameters γ, T1 and T2 are constants. Com-
bining the Friedmann equations with this more practical
EOS, we obtain the dynamical evolution equation for the
scale factor as
= −3γ − 2
. (35)
This model possesses a large variety of properties, such as
that we have found a scalar field model which is equiva-
lent to the above EOS. For related works on the modified
EOS, see Ref. [9, 11, 12, 13]. The present work can also
be regarded as a generalization of the EOS to γ = γ(a)
and T2 = T2(a). And the corresponding Hamiltonian
formalism for this system can be constructed similarly.
IV. RELATIONS TO THE OBSERVABLE
QUANTITIES
The observations of the supernovae (SNe) Ia have
provided the direct evidence for the cosmic accelerat-
ing expansion of our current Universe [1]. A bridge
between the cosmological theory and the observation
data is the H-z relation, where H ≡ ȧ/a is the Hub-
ble parameter and z is the redshift. For example, the
ΛCDM model in cosmology can be described mainly as
H2(z) = H20 [Ωm(1+z)
3+1−Ωm], where Ωm is the mat-
ter energy density. This model fits the observational data
well and provides the cosmological constant as the sim-
plest candidate for dark energy. In a sense, the different
cosmological models are characterized by the correspond-
ing H-z relations.
There is also a systematic way to construct the Hamil-
tonian starting from the general model
H2 = f(a), (36)
where f(a) is a specific function of the scale factor a,
according to the model. By differentiating Eq. (36), we
obtain that it is a solution of the following equation:
3γ − 2
3γf(a)
af ′(a)
, (37)
which possesses the same form of Eq. (5) or (10). The
corresponding coefficients are given by
f1(a) = −
3γ − 2
, f2(a) =
3γaf(a)
a2f ′(a)
. (38)
Then by applying Eq. (11) we can obtain the correspond-
ing Hamiltonian. Therefore, even if the EOS for a cosmo-
logical model is not explicitly linear in ρ, the Hamiltonian
formalism in the present work can also be applied if the
effective Friedmann equation H2 = f(a) can be given out
for that model.
Many approaches such as modified gravity [14] can
be reduced to effective Friedmann equations in the form
H2 = f(a). Since ΛCDM model fits the SNe Ia data well,
the reasonable cosmological models should be reduced to
Friedmann cosmology in an effective way and give out the
right H-z relation, in order to make a comparison with
the ΛCDM model. In our case, the Friedmann equations
in terms of the Hubble parameter can be written as
= −3γ
H2 + Λ̃(a). (39)
Here γ is assumed to be constant for simplicity. This
equation is linear in H2 and the effective term Λ̃(a) is
an inhomogeneous term. The solution in terms of H(z)
concerning the initial condition H(0) = H0 is given out
H(z)2 = H20 (1 + z)
Λ̃(z′)(1 + z′)−3γ−1dz′
In the power-law ΛCDM model, the contributions of dif-
ferent components are separated in H2, such as a con-
stant for the cosmological constant, and a (1 + z)2 fac-
tor for the curvature term. But in the general case, the
contribution of the matter cannot be separated from the
above solution. This problem is related to the conserva-
tion law of the matter, which has been investigated in
Refs. [5, 6].
The Bianchi identity for the energy-momentum tensor
Eq. (2) gives
ρ̇Λ + ρ̇+ 3H(ρ+ p) = 0, (41)
which implies that energy transfer will exist between the
matter and the vacuum energy. An intuitive idea has
been proposed that if both G and Λ are variable, the
ordinary energy-momentum tensor can be individually
conserved, i.e., ρ̇+3H(ρ+p) = 0 [6]. This is achieved by
combining the Bianchi identity for the variable G and Λ
model
[G(ρΛ + ρ)] + 3GH(ρ+ p) = 0, (42)
with the following constraint:
(ρ+ ρΛ)Ġ+Gρ̇Λ = 0. (43)
The authors of Ref. [6] assume that both the Newton con-
stant G and the cosmological constant Λ are functions of
a scale parameter µ and apply the renormalization group
approach to cosmology. If G(µ) evolves by a logarithmic
law and ρΛ(µ) evolves quadratically with µ, then this
picture can explain the evolution of the Universe, and at
the same time, the variable G can explain the flat rota-
tion curves of the galaxies without introducing the dark
matter hypothesis.
V. REMARKS ON QUANTUM COSMOLOGY
We have obtained a classical Hamiltonian formalism
of the Friedmann equations. Generally, once a Hamilto-
nian is obtained, the system can be quantized straightfor-
wardly by replacing the Poisson bracket with the commu-
tation relation [q, p] = i. However, we need to take into
account the ambiguity in the ordering of noncommuting
operators q and p. For simplicity, we ignore the order-
ing ambiguity here. In terms of the new variable φ, the
corresponding Schrödinger’s equation can be written as
H(φ, p̂φ)Ψ(φ) = EΨ(φ), (44)
where p̂φ = −i∂φ. To make a comparison between our ap-
proach and the Wheeler-DeWitt equation, we only take
the ΛCDM model as a very special case for an illustrative
example. The corresponding Hamiltonian for Eq. (19) in
the case γ = 1 is
H = 1
p2 − 1
Λa3, (45)
where p = aȧ. In the approach of the Wheeler-DeWitt
equation, H = 0 is a constraint [2, 15], thus the quanti-
zation gives (∂2a +
a4)Ψ(a) = 0. This is an anharmonic
oscillator with zero energy eigenvalue. In our case, the
Hamiltonian is nonzero and proportional to the matter
energy density, which we show in the following. The so-
lution of Eq. (39) with Λ̃(a) = Λ/2 is
H20 −
a−3 +
= H20 [Ωma
−3 + 1− Ωm], (46)
where Ωm ≡ 1 − Λ/(3H20 ). Therefore, the Hamiltonian
can be calculated as
H = a
H20Ωm. (47)
After a canonical transformation by Eq. (16), the
Schrödinger’s equation in terms of φ becomes
Ψ(φ) = EΨ(φ). (48)
Thus, for the asymptotic de Sitter Universe, the ΛCDM
model corresponds to an upside-down harmonic oscillator
in our formalism. Such an oscillator also appears in the
matrix description of de Sitter gravity [16].
We can transform the de Sitter Universe to the dual
anti-de Sitter Universe by employing the scale factor du-
ality [17], which has been found that a → a−1 gives
H → −H and other consequences. The duality for
Eq. (5) is given by
a → a−1, γ → −γ, Λ → −Λ, φ → −φ. (49)
It can be checked easily that Eq. (5) is invariant under
these transformations. If we use the dual scale factor a−1,
the corresponding potential becomes Ṽ (φ) = + 3
γ2Λφ2.
In fact, quantization in de Sitter spacetime is one of the
major difficulties of string theory at one time (though
this picture has changed a little bit after Kachru-Kallosh-
Linde- Trivedi theory appeared). It seems that quantiz-
ing de Sitter cosmology is no difference, since the time
variable used is the same, and it is known that there is no
global timelike coordinates in de Sitter spacetime. Some
quantum effects of a scalar field in de Sitter background
can be found in Ref. [18].
VI. CONCLUSION AND DISCUSSION
We have proposed a systematic scheme to describe the
Friedmann equations through a Hamiltonian formalism.
The generalized FRW model accompanied by both vari-
able EOS parameter and variable cosmological constant
admits a Hamiltonian description without constraint. Af-
ter an appropriate canonical transformation, the system
can be significantly simplified to an object moving in
an effective potential field. The bulk viscosity can also
be taken into account by a time-dependent Hamiltonian.
Some examples are given explicitly, such as the ΛCDM
model, the curvature term effect, and the inflation period.
The quantization of the system provides a new approach
to study the potential quantum cosmology, which is an
intriguing topic in theoretical physics research.
We shall discuss some possible future developments of
our work. As we have claimed, the formalism in this work
can be applied to a large variety of cosmological models.
By solving the Schrödinger equation H(φ, p̂φ)Ψ = EΨ,
the cosmological wave function can be obtained for a
specific model. Here we consider the curvature effect,
for example, which is described by the potential Eq. (21)
with parameters Λ = 0, γ = 1, and m = 2. The cor-
responding Schrödinger equation can be solved in terms
of the biconfluent Heun equation (BHE) [19]. We can
also start from the effective Lagrangian and study the
observational effects when we modify the potential. We
believe that our formalism would give a new perspective
to the potential study of quantum cosmology physics.
ACKNOWLEDGMENTS
J.R. thanks Prof. M.L. Ge for helpful discussions on
Hamiltonian systems. X.H.M. is supported by NSFC un-
der No. 10675062 and BK21 Foundation. L.Z. is sup-
ported by NSFC under No. 90403014.
APPENDIX A: MATHEMATICAL NOTES
A more general correspondence between a Hamiltonian
and its equation of motion is given in Ref. [19]. The
equation of motion of the Hamiltonian
H(q, p, t) =
P0(q, t)p
2 + P1(q, t)p+ P2(q, t)
is given out by
∂ lnP0
∂ ln f
− ∂ lnP0
− 2∂V
. (A2)
In the mathematical aspect, Eq. (28) can be further gen-
eralized to the following equation:
q̈ = F1(q, t)q̇
2 + F2(q, t)q̇ + F3(q, t), (A3)
however, here the coefficients F1 and F2 are not com-
pletely independent. Comparing with Eq. (A2), we can
see that the condition 2∂tF1(q, t) = ∂qF2(q, t) must be
satisfied for consistency. In the present work, both f1(q)
and η have safely satisfied this condition.
We shall explain why we choose the transformation as
in Eq. (12). Starting from the following equation
q̈ = f1(q)q̇
2 + ηq̇ + f2(q), (A4)
we expect that after an appropriate change of variable
φ(q), the above equation can be transformed as
φ̈ = ηφ̇+ g(φ). (A5)
By differentiating φ(q), we obtain φ̇ = φ′q̇, and φ̈ =
φ′′q̇2 + φ′q̈, where the prime denotes a derivative with
respect to q. Substituting φ, φ̇, and φ̈ into Eq. (A5), we
obtain
q̈ = −φ
q̇2 + ηq̇ +
Now it turns out that by defining −φ′′/φ′ = f1(q), which
can be solved as the form Eq. (12), the q̇2 term can be
eliminated.
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http://arxiv.org/abs/gr-qc/0612026

  We propose a Hamiltonian formalism for a generalized
Friedmann-Roberson-Walker cosmology model in the presence of both a variable
equation of state (EOS) parameter $w(a)$ and a variable cosmological constant
$\Lambda(a)$, where $a$ is the scale factor. This Hamiltonian system containing
1 degree of freedom and without constraint, gives Friedmann equations as the
equation of motion, which describes a mechanical system with a variable mass
object moving in a potential field. After an appropriate transformation of the
scale factor, this system can be further simplified to an object with constant
mass moving in an effective potential field. In this framework, the $\Lambda$
cold dark matter model as the current standard model of cosmology corresponds
to a harmonic oscillator. We further generalize this formalism to take into
account the bulk viscosity and other cases. The Hamiltonian can be quantized
straightforwardly, but this is different from the approach of the
Wheeler-DeWitt equation in quantum cosmology.

Hamiltonian formalism in Friedmann cosmology and its quantization
Jie Ren1,∗ Xin-He Meng2,3, and Liu Zhao2
Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin 300071, China
Department of physics, Nankai University, Tianjin 300071, China and
BK21 Division of Advanced Research and Education in physics, Hanyang University, Seoul 133-791, Korea
(Dated: October 15, 2018)
We propose a Hamiltonian formalism for a generalized Friedmann-Roberson-Walker cosmology
model in the presence of both a variable equation of state (EOS) parameter w(a) and a variable
cosmological constant Λ(a), where a is the scale factor. This Hamiltonian system containing 1
degree of freedom and without constraint, gives Friedmann equations as the equation of motion,
which describes a mechanical system with a variable mass object moving in a potential field. After
an appropriate transformation of the scale factor, this system can be further simplified to an object
with constant mass moving in an effective potential field. In this framework, the Λ cold dark matter
model as the current standard model of cosmology corresponds to a harmonic oscillator. We further
generalize this formalism to take into account the bulk viscosity and other cases. The Hamiltonian
can be quantized straightforwardly, but this is different from the approach of the Wheeler-DeWitt
equation in quantum cosmology.
PACS numbers: 98.80.Jk,45.20.Jj,03.50.-z
I. INTRODUCTION
Since the current accelerating expansion of our Uni-
verse was discovered [1] around 1998 and 1999, theoret-
ical physicists have devoted increasingly more attention
to the Friedmann-Roberson-Walker (FRW) model as a
standard framework in cosmology study. The Λ cold dark
matter (ΛCDM) model as the standard model of cosmol-
ogy so far fits well with observational data whereas it
has had some serious theoretical problems. To make a
comparison to the ΛCDM model, physicists have built
many cosmological models that are able to give out the
effective Friedmann equations with variable cosmological
constant. To quantize the Friedmann equations, the com-
monly used theory is the Wheeler-DeWitt equation [2],
which has been studied and applied widely in quantum
cosmology [3]. Starting from the Hilbert-Einstein action
with the Roberson-Walker (RW) metric, the correspond-
ing HamiltonianH can be obtained. Then the Friedmann
equation plays the role as the constraint H = 0, which
leads to the Wheeler-DeWitt equation. In the present
work, we consider the Friedmann equations as basic equa-
tions and find a Hamiltonian system that gives Fried-
mann equations as classical equations of motion without
constraint.
Many Ansätze of the variable cosmological constant
have been studied in the literature, such as Refs. [4, 5,
6, 7]. Moreover, some models motivated from the string
theory give an effective cosmological term when reduced
to the FRW framework. We assume that the equation
of state (EOS) parameter w ≡ p/ρ can also be variable,
which means that the contents of the Universe, except
the cosmological term, are generalized to a nonperfect
∗Electronic address: jrenphysics@hotmail.com
fluid, or perfect fluid as a special case. In observational
cosmology, the redshift z is regarded as an observable
quantity and related to the scale factor a by z ≡ a0/a−1.
Therefore, we investigate a general case that both the
EOS parameter w and the cosmological constant Λ can
be functions of the scale factor a, and take into account
the bulk viscosity.
As an extension of the problem, we construct a Hamil-
tonian formalism for a system described by the following
equation:
q̈ = f1(q)q̇
2 + ηq̇ + f2(q),
where f1(q) and f2(q) are arbitrary functions, and η is
constant. Also it can be regarded as a generalization
of the damping harmonic oscillator. The corresponding
Hamiltonian describes an object with variable mass mov-
ing in a potential field. After an appropriate canonical
transformation, this system can be further simplified to
an object with constant mass moving in an effective po-
tential field. Thus, differential models in the FRW frame-
work are characterized by their effective potentials. This
is a general formalism and it can be applied to many cos-
mological models, for example, that the ΛCDM model
corresponds to a harmonic oscillator. Since the quanti-
zation of Friedmann equations can provide an insight to
quantum cosmology as a glimpse of quantum gravity, we
also make some remarks on the quantum case, which pro-
vides a correspondence between cosmology and quantum
mechanics.
The paper is organized as follows. In Sec. II we present
a generalized FRW model and the corresponding Hamil-
tonian to describe the Friedmann equations. Then we
find a canonical transformation to further simplify the
problem, and give some examples and special cases. In
Sec. III we show that our framework can also be applied
in the dissipative case with bulk viscosity. In Sec. IV
we turn our attention to the relation to the observable
http://arxiv.org/abs/0704.0672v3
mailto:jrenphysics@hotmail.com
quantities and review some issues of the Bianchi identity.
In Sec. V we make some remarks on quantum cosmology
from our approach. In the last section we present the
conclusion and discuss some future subjects.
II. HAMILTONIAN FORMALISM
A. Hamiltonian description of the Friedmann
equations
We consider the RW metric in the flat space geometry
(k=0) as the case favored by current cosmic observational
data:
ds2 = −dt2 + a(t)2(dr2 + r2dΩ2), (1)
where a(t) is the scale factor. The energy-momentum
tensor for the cosmic fluid can be written as
T̃µν = (ρ+ p)UµUν + (p+ ρΛ)gµν , (2)
where ρΛ = Λ/(8πG) is the energy density of the cos-
mological constant. Thus, Einstein’s equation Rµν −
gµνR = 8πGT̃µν contains two independent equations:
, (3a)
= −4πG
(ρ+ 3p) +
. (3b)
The EOS of the matter (cosmic fluid except the cosmo-
logical constant) is commonly assumed to be
p = (γ − 1)ρ. (4)
Cosmologists usually call Eq. (3a) as the Friedmann
equation and Eq. (3b) as the acceleration equation in the
literature, whereas for simplicity we name both Eqs. (3a)
and (3b) Friedmann equations here. For generality, we
assume that both γ and Λ are functions of the scale factor
a, thus we call it the generalized FRWmodel. Combining
the Friedmann equations with the EOS, we obtain
= −3γ(a)− 2
γ(a)Λ(a)
, (5)
which determines the evolution of the scale factor.
We regard Eq. (5) as a basic starting point; therefore, if
the dynamical equation for the scale factor can be written
as that form, the present framework can be valid. If
the Newton constant G is constant and the cosmological
constant Λ is variable, the energy-momentum tensor for
the matter cannot individually conserved [5, 6], which
implies an interaction between the matter and vacuum
energy. In the following, we assume G to be constant
until Sec. IV.
Our aim is to find a Hamiltonian description of Eq. (5)
as the classical equation of motion. We start from the
following Lagrangian
L(q, q̇) = 1
M(q)q̇2 − V (q), (6)
and the corresponding Hamiltonian thus is
H(q, p) = p
2M(q)
+ V (q), (7)
with the canonical Poisson bracket {q, p} = 1. One can
check that the equation of motion for Eq. (6) or (7) is
∂ lnM
∂ ln q
. (8)
This equation possesses the same form as Eq. (5). There-
fore, by comparing Eq. (5) with Eq. (8), we can take a as
the general coordinate and solve the functions M(a) and
V (a). Then the Lagrangian L = 1
M(a)q̇2 − V (a) with
M = exp
3γ − 2
, V = −1
MγΛada, (9)
gives Eq. (5) as the equation of motion. For some spec-
ified functions γ = γ(a) and Λ = Λ(a), the above in-
tegrations can be evaluated out to give M(a) and V (a)
explicitly.
Now we can see that the generalized FRWmodel essen-
tially corresponds to an object with variable mass M(a)
moving in a potential field V (a). In the following, we
will show that this picture can be further simplified as
an object with constant mass moving in an effective po-
tential field Ṽ (φ), after an appropriate transformation of
the scale factor.
B. Canonical transformation
The above problem can be generalized as the Hamilto-
nian description of the nonlinear equation
q̈ = f1(q)q̇
2 + f2(q), (10)
where f1(q) and f2(q) are two specified functions.
This equation can be derived by the Lagrangian L =
M(q)q̇2 − V (q) with
M = exp
f1(q)dq
, V = −
Mf2(q)dq. (11)
We define a new variable φ as (see Appendix)
f1(q)dq
dq. (12)
This transformation can eliminate the q̇2 term and gives
the equation for the variable φ as in
φ̈ = f2(q) exp
f1(q)dq
, (13)
where q → φ denotes using Eq. (12) to change the vari-
able q to φ. Since there is no φ̇2 term in Eq. (13), this can
be regarded as a partial linearization. Therefore, the sys-
tem of Eq. (10) transformed to Eq. (13) can be described
by the Lagrangian
L(φ, φ̇) = 1
φ̇2 − Ṽ (φ), (14)
with the potential as
Ṽ (φ) = −
f2(q) exp
f1(q)dq
f2(q) exp
f1(q)dq
.(15)
The simplification of the problem by Eq. (12) is essen-
tially the canonical transformation
q → φ, pq → pφ, H(q, pq) → H(φ, pφ), (16)
where pq = M(q)q̇, pφ = φ̇, and H(φ, pφ) = 12p
φ + Ṽ (φ).
Therefore, the classical and quantum properties of differ-
ent models are characterized by the effective potentials.
For Eq. (5) as a special case, the new variable φ is
given by
3γ − 2
da. (17)
C. Some examples
We will give some special cases of the above general
framework to show some applications. If both γ and Λ
are constant for a simple case, the integrations in Eq. (17)
can be evaluated out as
a3γ/2, γ 6= 0, (18a)
= ln a, γ = 0. (18b)
Now we consider γ 6= 0 for example. The special case γ =
1 corresponds to the ΛCDM model. The equation for φ
can be obtained as φ̈− 3
γ2Λφ = 0, and the corresponding
Lagrangian is
L = 1
φ̇2 +
γ2Λφ2. (19)
We can see that the simplest model in cosmology just
corresponds to a harmonic oscillator after linearization.
In particular, this is a upside-down harmonic oscillator
for the asymptotic de Sitter Universe.
We can add the curvature effect to the ΛCDM model,
which is described by the special case m = 2 of the fol-
lowing equation:
= −3γ − 2
. (20)
Here the parameters γ, Λ, and m are all constants. This
equation possesses the same form of Eq. (10). By defining
φ as Eq. (12) and using Eq. (15), we obtain the effective
potential as
Ṽ (φ) = −3
γ2Λφ2 +
3γ −m
)2−2m/3γ
, (21)
for γ 6= 0 and m 6= 3γ.
Another example is the Friedmann equations during
the inflation era. In the study of inflation, we usually
use the conformal time τ instead of the comoving cosmic
time t. Here we assume that a constant term −p0 is in
the EOS during inflation. Thus the Friedmann equations
combined with the EOS p = −ρ− p0 yield
p0, (22)
where the prime denotes a derivative with respect to τ ,
and κ2 = 8πG. By defining φ = −1/a, the equation for
φ is φ′′φ+ (κ2/2)p0 = 0. The effective potential is thus
Ṽ (φ) =
p0 ln |φ|. (23)
Moreover, if we add the curvature term in this case, it
corresponds to a φ2 potential.
III. BULK VISCOSITY
We assume that the cosmic fluid possesses some dis-
sipation effects. Since the sheer tensor σµν = 0 for RW
metric, the sheer viscosity does not contribute to the evo-
lution in Friedmann cosmology. The energy-momentum
tensor for nonperfect fluid concerning bulk viscosity in
the right-hand side of Einstein’s equation is given by [8, 9]
Tµν = ρUµUν + (p− ζ0θ)hµν , (24)
where hµν ≡ gµν + UµUν is the projection operator,
θ ≡ Uα;α = 3ȧ/a is the scalar expansion, and ζ is the
bulk viscosity coefficient. Consequently, Eq. (5) should
be modified as
= −3γ(a)− 2
+ 12πGζ0
γ(a)Λ(a)
. (25)
where both γ and Λ can be functions of a for generality,
and ζ0 is constant. We also find a Hamiltonian
H(a, pa, t) =
2M(a, t)
+ V (a, t), (26)
with the Poisson bracket {a, pa} = 1 to give Eq. (25) as
the classical equation of motion. The functions in this
Hamiltonian are given by
M = exp
3γ − 2
da− 12πGζ0t
, (27a)
V = −1
MγΛada. (27b)
Although a dissipative system cannot be described by
a conservative Hamiltonian generally, one can directly
check that the classical equation of motion for the Hamil-
tonian Eq. (26) is Eq. (25). As a special case, the equa-
tion for a damping harmonic oscillator can be derived by
the Caldirora-Kani (CK) Hamiltonian [10].
The above problem can be generalized to construct a
Hamiltonian system for the equation
q̈ = f1(q)q̇
2 + ηq̇ + f2(q), (28)
where η is constant. It can be derived by the Hamiltonian
H(q, p, t) = 1
M(q, t)−1p2 + V (q, t) with
M = exp
f1(q)dq − ηt
, V = −
Mf2(q)dq.
Similarly, by using the new variable φ defined by
Eq. (12), the equation for φ is
φ̈ = ηφ̇+ f2(q) exp
f1(q)dq
. (30)
Now we consider a very special case that both γ and Λ
are constant; then φ defined by Eq. (18a) satisfies
φ̈− 12πGζ0φ̇−
γ2Λφ = 0, (31)
which describes a damping harmonic oscillator.
The damping harmonic oscillator
Mq̈ = −ηq̇(t)−
∂V (q)
, (32)
has been studied in quantum mechanics. The CK Hamil-
tonian
e−ηt/Mp2 +
Mω2eηt/M q2, (33)
with the commutation relation [q, p] = i~, can yield the
dissipation equation (32) through the Heisenberg equa-
tion [10]. Our work can be regarded as a generalization
to the case of variable mass. It is the variable mass that
generates a nonlinear term in the equation of motion that
describes the generalized FRW model.
In our previous work [9], we have proposed an EOS as
p = (γ − 1)ρ−
κ2T 22
where the parameters γ, T1 and T2 are constants. Com-
bining the Friedmann equations with this more practical
EOS, we obtain the dynamical evolution equation for the
scale factor as
= −3γ − 2
. (35)
This model possesses a large variety of properties, such as
that we have found a scalar field model which is equiva-
lent to the above EOS. For related works on the modified
EOS, see Ref. [9, 11, 12, 13]. The present work can also
be regarded as a generalization of the EOS to γ = γ(a)
and T2 = T2(a). And the corresponding Hamiltonian
formalism for this system can be constructed similarly.
IV. RELATIONS TO THE OBSERVABLE
QUANTITIES
The observations of the supernovae (SNe) Ia have
provided the direct evidence for the cosmic accelerat-
ing expansion of our current Universe [1]. A bridge
between the cosmological theory and the observation
data is the H-z relation, where H ≡ ȧ/a is the Hub-
ble parameter and z is the redshift. For example, the
ΛCDM model in cosmology can be described mainly as
H2(z) = H20 [Ωm(1+z)
3+1−Ωm], where Ωm is the mat-
ter energy density. This model fits the observational data
well and provides the cosmological constant as the sim-
plest candidate for dark energy. In a sense, the different
cosmological models are characterized by the correspond-
ing H-z relations.
There is also a systematic way to construct the Hamil-
tonian starting from the general model
H2 = f(a), (36)
where f(a) is a specific function of the scale factor a,
according to the model. By differentiating Eq. (36), we
obtain that it is a solution of the following equation:
3γ − 2
3γf(a)
af ′(a)
, (37)
which possesses the same form of Eq. (5) or (10). The
corresponding coefficients are given by
f1(a) = −
3γ − 2
, f2(a) =
3γaf(a)
a2f ′(a)
. (38)
Then by applying Eq. (11) we can obtain the correspond-
ing Hamiltonian. Therefore, even if the EOS for a cosmo-
logical model is not explicitly linear in ρ, the Hamiltonian
formalism in the present work can also be applied if the
effective Friedmann equation H2 = f(a) can be given out
for that model.
Many approaches such as modified gravity [14] can
be reduced to effective Friedmann equations in the form
H2 = f(a). Since ΛCDM model fits the SNe Ia data well,
the reasonable cosmological models should be reduced to
Friedmann cosmology in an effective way and give out the
right H-z relation, in order to make a comparison with
the ΛCDM model. In our case, the Friedmann equations
in terms of the Hubble parameter can be written as
= −3γ
H2 + Λ̃(a). (39)
Here γ is assumed to be constant for simplicity. This
equation is linear in H2 and the effective term Λ̃(a) is
an inhomogeneous term. The solution in terms of H(z)
concerning the initial condition H(0) = H0 is given out
H(z)2 = H20 (1 + z)
Λ̃(z′)(1 + z′)−3γ−1dz′
In the power-law ΛCDM model, the contributions of dif-
ferent components are separated in H2, such as a con-
stant for the cosmological constant, and a (1 + z)2 fac-
tor for the curvature term. But in the general case, the
contribution of the matter cannot be separated from the
above solution. This problem is related to the conserva-
tion law of the matter, which has been investigated in
Refs. [5, 6].
The Bianchi identity for the energy-momentum tensor
Eq. (2) gives
ρ̇Λ + ρ̇+ 3H(ρ+ p) = 0, (41)
which implies that energy transfer will exist between the
matter and the vacuum energy. An intuitive idea has
been proposed that if both G and Λ are variable, the
ordinary energy-momentum tensor can be individually
conserved, i.e., ρ̇+3H(ρ+p) = 0 [6]. This is achieved by
combining the Bianchi identity for the variable G and Λ
model
[G(ρΛ + ρ)] + 3GH(ρ+ p) = 0, (42)
with the following constraint:
(ρ+ ρΛ)Ġ+Gρ̇Λ = 0. (43)
The authors of Ref. [6] assume that both the Newton con-
stant G and the cosmological constant Λ are functions of
a scale parameter µ and apply the renormalization group
approach to cosmology. If G(µ) evolves by a logarithmic
law and ρΛ(µ) evolves quadratically with µ, then this
picture can explain the evolution of the Universe, and at
the same time, the variable G can explain the flat rota-
tion curves of the galaxies without introducing the dark
matter hypothesis.
V. REMARKS ON QUANTUM COSMOLOGY
We have obtained a classical Hamiltonian formalism
of the Friedmann equations. Generally, once a Hamilto-
nian is obtained, the system can be quantized straightfor-
wardly by replacing the Poisson bracket with the commu-
tation relation [q, p] = i. However, we need to take into
account the ambiguity in the ordering of noncommuting
operators q and p. For simplicity, we ignore the order-
ing ambiguity here. In terms of the new variable φ, the
corresponding Schrödinger’s equation can be written as
H(φ, p̂φ)Ψ(φ) = EΨ(φ), (44)
where p̂φ = −i∂φ. To make a comparison between our ap-
proach and the Wheeler-DeWitt equation, we only take
the ΛCDM model as a very special case for an illustrative
example. The corresponding Hamiltonian for Eq. (19) in
the case γ = 1 is
H = 1
p2 − 1
Λa3, (45)
where p = aȧ. In the approach of the Wheeler-DeWitt
equation, H = 0 is a constraint [2, 15], thus the quanti-
zation gives (∂2a +
a4)Ψ(a) = 0. This is an anharmonic
oscillator with zero energy eigenvalue. In our case, the
Hamiltonian is nonzero and proportional to the matter
energy density, which we show in the following. The so-
lution of Eq. (39) with Λ̃(a) = Λ/2 is
H20 −
a−3 +
= H20 [Ωma
−3 + 1− Ωm], (46)
where Ωm ≡ 1 − Λ/(3H20 ). Therefore, the Hamiltonian
can be calculated as
H = a
H20Ωm. (47)
After a canonical transformation by Eq. (16), the
Schrödinger’s equation in terms of φ becomes
Ψ(φ) = EΨ(φ). (48)
Thus, for the asymptotic de Sitter Universe, the ΛCDM
model corresponds to an upside-down harmonic oscillator
in our formalism. Such an oscillator also appears in the
matrix description of de Sitter gravity [16].
We can transform the de Sitter Universe to the dual
anti-de Sitter Universe by employing the scale factor du-
ality [17], which has been found that a → a−1 gives
H → −H and other consequences. The duality for
Eq. (5) is given by
a → a−1, γ → −γ, Λ → −Λ, φ → −φ. (49)
It can be checked easily that Eq. (5) is invariant under
these transformations. If we use the dual scale factor a−1,
the corresponding potential becomes Ṽ (φ) = + 3
γ2Λφ2.
In fact, quantization in de Sitter spacetime is one of the
major difficulties of string theory at one time (though
this picture has changed a little bit after Kachru-Kallosh-
Linde- Trivedi theory appeared). It seems that quantiz-
ing de Sitter cosmology is no difference, since the time
variable used is the same, and it is known that there is no
global timelike coordinates in de Sitter spacetime. Some
quantum effects of a scalar field in de Sitter background
can be found in Ref. [18].
VI. CONCLUSION AND DISCUSSION
We have proposed a systematic scheme to describe the
Friedmann equations through a Hamiltonian formalism.
The generalized FRW model accompanied by both vari-
able EOS parameter and variable cosmological constant
admits a Hamiltonian description without constraint. Af-
ter an appropriate canonical transformation, the system
can be significantly simplified to an object moving in
an effective potential field. The bulk viscosity can also
be taken into account by a time-dependent Hamiltonian.
Some examples are given explicitly, such as the ΛCDM
model, the curvature term effect, and the inflation period.
The quantization of the system provides a new approach
to study the potential quantum cosmology, which is an
intriguing topic in theoretical physics research.
We shall discuss some possible future developments of
our work. As we have claimed, the formalism in this work
can be applied to a large variety of cosmological models.
By solving the Schrödinger equation H(φ, p̂φ)Ψ = EΨ,
the cosmological wave function can be obtained for a
specific model. Here we consider the curvature effect,
for example, which is described by the potential Eq. (21)
with parameters Λ = 0, γ = 1, and m = 2. The cor-
responding Schrödinger equation can be solved in terms
of the biconfluent Heun equation (BHE) [19]. We can
also start from the effective Lagrangian and study the
observational effects when we modify the potential. We
believe that our formalism would give a new perspective
to the potential study of quantum cosmology physics.
ACKNOWLEDGMENTS
J.R. thanks Prof. M.L. Ge for helpful discussions on
Hamiltonian systems. X.H.M. is supported by NSFC un-
der No. 10675062 and BK21 Foundation. L.Z. is sup-
ported by NSFC under No. 90403014.
APPENDIX A: MATHEMATICAL NOTES
A more general correspondence between a Hamiltonian
and its equation of motion is given in Ref. [19]. The
equation of motion of the Hamiltonian
H(q, p, t) =
P0(q, t)p
2 + P1(q, t)p+ P2(q, t)
is given out by
∂ lnP0
∂ ln f
− ∂ lnP0
− 2∂V
. (A2)
In the mathematical aspect, Eq. (28) can be further gen-
eralized to the following equation:
q̈ = F1(q, t)q̇
2 + F2(q, t)q̇ + F3(q, t), (A3)
however, here the coefficients F1 and F2 are not com-
pletely independent. Comparing with Eq. (A2), we can
see that the condition 2∂tF1(q, t) = ∂qF2(q, t) must be
satisfied for consistency. In the present work, both f1(q)
and η have safely satisfied this condition.
We shall explain why we choose the transformation as
in Eq. (12). Starting from the following equation
q̈ = f1(q)q̇
2 + ηq̇ + f2(q), (A4)
we expect that after an appropriate change of variable
φ(q), the above equation can be transformed as
φ̈ = ηφ̇+ g(φ). (A5)
By differentiating φ(q), we obtain φ̇ = φ′q̇, and φ̈ =
φ′′q̇2 + φ′q̈, where the prime denotes a derivative with
respect to q. Substituting φ, φ̇, and φ̈ into Eq. (A5), we
obtain
q̈ = −φ
q̇2 + ηq̇ +
Now it turns out that by defining −φ′′/φ′ = f1(q), which
can be solved as the form Eq. (12), the q̇2 term can be
eliminated.
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