Pis’ma v ZhETF, vol. 111, iss. 8, pp. 485 - 486

April 25

Generalized unimodular gravity in Friedmann and Kantowski-Sachs

universes

A. Yu. Kamenshchik^+∗1), A. Tronconi⁺, G. Venturi⁺

⁺Dipartimento di Fisica e Astronomia, Università di Bologna and Istituto Nazionale di Fisica Nucleare,

via Irnerio 46, 40126 Bologna, Italy

^∗L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117940 Moscow, Russia

Submitted 19 March 2020

Resubmitted 23 March 2020

Accepted 23

March 2020

DOI: 10.31857/S1234567820080029

One of the oldest modified theories of gravity is uni-

ary model based on generalized unimodular gravity and

modular gravity, dating back to the paper by Einstein

the behaviour of linear perturbations in this model was

[1]. The recent rebirth of this idea is connected with pa-

studied.

pers [2, 3]. The main point of unimodular gravity con-

However, the model [5] opens some interesting op-

sists of the fact that when one requires that the deter-

portunities already at the level of a simple minisuper-

minant of the metric is fixed, the cosmological constant

space models with finite number of degrees of freedom.

arises as an integration constant in the Einstein equa-

We shall discuss here some of them. For a flat Friedmann

tions. The unimodular gravity theories can essentially

model with the metric ds² = -N²(t)dt² + a²(t)dl², γ =

be generalized by using the Arnowitt Deser-Misner

= a⁶ and the equation of state is simply w = 1 dlnN(a)3dlna.

(ADM) [4] approach to gravity.

One can derive this equation directly from the Fried-

Such a generalization was suggested recently in pa-

mann model. The Lagrangian for the flat Friedmann

a²a

per [5]. If one treats the lapse function N not as La-

universe without matter can be written as L =

grange multiplier, giving one of the constraints of the

If we now treat the lapse function as a function of the

theory, but as a given function of the determinant of

scale factor a, the variation with respect to a gives the

the spatial metric γ, then in the equations of motion an

following Euler-Lagrange equation: 2^äaN + a^2d(a/N)da = 0,

effective matter arises with the equation of state param-

where the “dot” signifies the differentiation with respect

eter w given by w = 2^{dlnN(γ)dlnγ} .

to the time parameter t. This equation has the first in-

a²a

Thus, on treating one of the Lagrange multipliers of

tegral

= C, where C is a constant. Dividing this

the General Relativity, i.e., the lapse function N not as

equation by Na³, we obtain

a Lagrange multiplier, but as a given function of other

a²

( da)2

variables, we freeze one of the symmetries of the system

(1)

N²a²

a²

dτ

Na³

and as a result the effective matter content of the theory

becomes richer. This phenomenon is quite well-known

where τ is the cosmic or synchronous time dτ = Ndt.

and was pioneered by Dirac in paper [6] dedicated to

This equation can be interpreted as the first Friedmann

electrodynamics.

equation for a flat universe filled with matter having

In spite of its simplicity the model of generalized

the energy density ε =CNa3 . On remembering the en-

unimodular gravity [5] imposes some interesting prob-

ergy conservation law

lems and opens some attractive prospects due to its un-

dε

ε+p

expected flexibility. In paper [7] the Hamiltonian for-

= -3

(2)

malism for this model, treated as a rather complicated

example of a constrained dynamical system [8], was con-

we can immediately find the pressure

sidered in detail. Especially interesting in this context

dε

C dN

1 dlnN

is the question of the determination of the number and

p=-

-ε=

ε,

the character of the physical degrees of freedom, aris-

3N²a² da

3 dlna

ing here. The paper [9] was devoted to the inflation-

which confirms the relation presented above.

It is known that the observed cosmic acceleration of

¹⁾e-mail: kamenshchik@bo.infn.it

the universe requires the presence of a so called dark

Письма в ЖЭТФ том 111 вып. 7 - 8

2020

485

486

A. Yu. Kamenshchik, A. Tronconi, G. Venturi

energy with negative pressure. Some observations indi-

scale factor b is b = b₀ tanh^t2 -^A . On using the duality_a

cate that the corresponding equation of state parameter

relations, we obtain the following Schwarzschild-type

is less than -1: w < -1. Such a kind of dark energy is

metric:

[

√

]

called “phantom dark energy”. The evolution in the pres-

(

a₀ )

Ab₀

a₀

A²

ence of such energy implies the future encounter with

ds² = b²

-2

dt² -

a₀

a²

a cosmological singularity called “Big Rip” [10, 11]. Its

dR²

scale factor and its time derivative tend to infinity.

- R²(dθ² + sin² θdφ²).

(6)

1-a0

However, one can imagine a less dramatic scenario

for the development of the universe, wherein the phan-

We see that while the spatial part of the metric has not

tom or super-acceleration stage is a temporary one. In

changed, the coefficient g₀₀ for dt² has changed essen-

this case the universe should pass through the phantom

tially. If the constant A was positive then the metric

divide line which means that the sign of the expression

coefficient vanishes at

w + 1 changes. We wish to show that, at least at the

a₀

R₀ =

>a₀,

level of the Friedmann model, the generalized unimod-

1-A2

a²⁰b²

ular gravity can easily describe the phantom divide line

crossing.

provided A² < a²⁰b²⁰. We should then think of how to

Indeed, it is enough to choose the lapse function as

describe the continuation of the metric into the region

follows:

where R < R₀ and then to R < a₀. If A is negative

N =

+ Fa.

(3)

(the energy density of the effective matter is negative)

a⁵

the expression for b cannot become equal to zero, but

On remaining in the field of minisuperspace mod-

we still stumble upon the problem of its behaviour for

els with a finite number of degrees of freedom, we can

R<a₀.

already suggest a further simple generalization of uni-

Full text of the paper is published in JETP Letters

modular gravity. In particular the lapse function can de-

journal. DOI: 10.1134/S0021364020080032

pend not on the determinant of the spatial metric, but

on some other combination of components of the spatial

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(5)

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The expression for a is now the same as before, while the

Письма в ЖЭТФ том 111 вып. 7 - 8

2020