Creation of wormholes during the cosmological bounce Petar Pavlović1and Marko Sossich21

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Creation of wormholes during the cosmological

bounce

Petar Pavlović ∗1and Marko Sossich †2,1

1Institute for Cosmology and Philosophy of Nature, Trg svetog Florijana 16,

Križevci, Croatia

2University of Zagreb, Faculty of Electrical Engineering and Computing, Department

of Physics, Unska 3, 10 000 Zagreb, Croatia

February 7, 2025

Abstract

In this work we demonstrate that wormholes can in principle be nat-

urally created during the cosmological bounce without the need for the

exotic matter or any kind of additional modiﬁcations of the gravitational

sector, apart from the one enabling the cosmological bounce. This re-

sult is general and does not depend on the details of the modiﬁcations of

gravitational equations needed to support the bounce. To study the possi-

ble existence of wormholes around the cosmological bounce we introduce

general modiﬁcations of Einstein’s ﬁeld equations need to support the

bouncing solutions. In this regime we show that it is possible to construct

a cosmological wormhole solution supported by matter, radiation and vac-

uum energy, satisfying the Weak Energy Condition (WEC), which asymp-

totically approaches the Friedmann-Lemaître-Robertson-Walker (FLRW)

metric. However, at a speciﬁc cosmological time, which depends on the

parameters of the bouncing cosmological model, the WEC describing the

matter needed to support such wormholes is spontaneously violated. This

means that such wormholes could potentially exist in large numbers during

some period around the bounce, signiﬁcantly changing the causal struc-

ture of space-time, and then vanish afterwards.

1 Introduction

Our current understanding of the cosmological evolution is based on Einstein’s

general theory of relativity, which is one of the most successful physical theories.

∗petar.pavlovic@kozmologija.org

†marko.sossich@fer.hr

arXiv:2210.06142v4 [gr-qc] 6 Feb 2025

General theory of relativity was so far veriﬁed by various types of experiments

– from the light deﬂection and the perihelion advance of Mercury to the recent

detection of gravitational waves [1, 2, 3, 4]. One of the consequences of this

theory is the necessary existence of singularities if the space-time is respecting

some usual causal properties and if the matter is respecting the usual energy

conditions (there are diﬀerent variants of this result including the Strong, Null

or Weak Energy Condition), as proven by the singularity theorems of Hawking

[5, 6, 7]. One consequence of this result is the necessary existence of singularity

in the evolution of our Universe, called the big bang singularity, if general theory

of relativity is correct. However, the physical relevance of this result is highly

questionable – since it is precisely in such strong gravity regimes that we should

doubt the validity of Einstein’s general relativity as the correct description of

gravity. First of all, for strong gravitational ﬁelds both quantum behaviour of

matter ﬁelds and space-time itself will probably become important and signiﬁ-

cantly change the ﬁeld equations for gravity. The proper understanding of such

regimes therefore requires a proper knowledge of quantum theory of gravity,

which is, of course, still currently not available. On the other hand, assum-

ing the actual physical existence of singularities would mean the capitulation

with the respect to the fundamental goal of physics – namely, the complete,

non-divergent and consistent description of reality, including the evolution of

the Universe. For all this reasons, we should view the Hawking singularity theo-

rems more as a signal of incompleteness of Einstein’s general theory of relativity,

than the proof for the actual physical existence of singularities. Furthermore,

we believe that the demand for singularity-free solutions constitutes one of the

most important criteria for the future quantum theory of gravity, and therefore

also for the eﬀective theories which are being investigated in order to overcome

the current gap between the quantum physics and description of gravity as a

geometry of space-time.

Various investigations in the past decades have demonstrated that even simple

modiﬁcations of gravitational Lagrangian with respect to the standard Einstein-

Hilbert action, while leaving all other physical assumptions of Einstein’s gen-

eral relativity intact, can prevent the appearance of the big-bang singularity

[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Also, in higher curvature

gravity theories some important non-singular investigations have been done in

Gauss-Bonnet higher curvature gravity [22, 23, 24] and in f(R)gravity theo-

ries [25, 26, 27]. In such models the big bang singularity is then replaced by

the cosmological bounce, in which the Universe undergoes a transition from

contraction to expansion. It is worth remembering that there are in principle

no real physical reasons for favoring Einstein-Hilbert action in comparison to

higher curvature modiﬁcations, such as f(R),f(T)or higher derivative gravity

theories, as long as they lead to the same observable weak ﬁeld limit and have

no theoretical pathologies. As a matter of fact, the Einstein-Hilbert action was

historically introduced as the simplest action leading to the Newtonian limit,

out of the inﬁnitely many other equally possible options. The investigation of

possible modiﬁcations of gravitational action, and their consequences on the

existence of the big bang singularity, are therefore important not just for trying

to overcome the limitations of general relativity - such as the existence of singu-

larities, but also as the path for better understanding the structure of potential

quantum theory of gravity.

There were also numerous works which demonstrated the possibility of the cos-

mological bounce, if some new hypothetical additions to the standard cosmo-

logical model are added, such as speciﬁc scalar ﬁelds, extra dimensions or new

types of ﬂuids [28, 29, 30, 31, 32, 33, 34, 35]. Although such investigations

can lead to some important insights regarding the problem of initial singularity

in the cosmological evolution, we think that, from a methodological point of

view, the approach based on the modiﬁcation of ﬁeld equations, assuming no

new ingredients, should be viewed as superior. This is because the addition of

new hypothetical structures and substances should be disfavored with respect

to explanation which assumes no new unobserved and yet unveriﬁed forms of

matter-energy or spacetime structure. To put it in diﬀerent words, it is always

possible to obtain the desired physical goal by increasing the number of param-

eters and invoking various types of ad hoc entities, but by doing so, the physical

theory looses its simplicity, necessity and integrity.

One of the important limitations of the usual strategy of investigating bouncing

cosmologies based on speciﬁc constructions (i.e. the speciﬁc type of modiﬁca-

tions of Einstein’s general relativity or matter-energy content of the Universe)

is that the obtained results are highly dependent on the speciﬁc assumptions

which are taken to derive them. It is thus not easy, and sometimes simply not

possible, to see which of the properties of solutions are general and which are

the result of highly hypothetical and often not properly motivated new modiﬁ-

cations and additions. This problem signiﬁes the need for a model independent

study of bouncing cosmologies. We ﬁrst tried to contribute to this research pro-

gram by studying the bouncing and cyclic solutions supported by a general type

of higher order curvature corrections [36]. This research was then extended and

generalized by studying the model independent dynamic properties of bouncing

cosmologies and then applying the results to diﬀerent types of modiﬁed gravity

theories [37]. It was also demonstrated that the problem of magnetogenesis has

a simple possible solution in the model-independent approach to bouncing cos-

mology [38]. Recently, we proposed a model-independent approach to bouncing

cosmology in which we proposed a simple solution of the cosmological constant

problem and also studied the eﬀects of quantum ﬂuctuations of the spacetime

geometry [39]. We will use some of the obtained conclusion in the present study

of wormholes created during the cosmological bounce.

There is an obvious technical connection between cosmological bounce and

another type of hypothetical gravitational solutions - wormholes, that comes

from the fact that both types of solutions imply physics beyond standard gen-

eral relativity. Wormholes are solutions of ﬁeld equations for gravity which

represent a shortcut through spacetime, a tube-like structure which is asymp-

totically ﬂat at both ends, and connects two distant parts of the Universe.

While such spacetime conﬁguration is actually a solution of classical Einstein’s

equations, it requires the violation of Weak Energy Condition (WEC) for its

existence in the framework of Einstein’s gravity [40]. However, it is at the

present time not known which kind of substance could lead to the needed vio-

lation of the usual WEC on macroscopic scales. Therefore, if WEC is assumed

to be satisﬁed then wormholes can be realized only by virtue of modiﬁcation

of ﬁeld equations for gravity. Numerous realizations of such wormhole solu-

tions not requiring the WEC violation were extensively studied in the literature

[41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60].

One special case of potential wormhole solutions supported by modiﬁed grav-

ity are “small cosmological wormholes”, an approximate type of solution where

for a large enough distance from the wormhole throat the spacetime geometry

can be described by the standard cosmological FLRW metric [61]. Such type

of solutions enable us to simply study wormholes which are contained in the

expanding Universe.

As we discussed, both the bouncing cosmological solutions and wormholes can

be supported by an eﬀective violation of usual energy conditions coming from

the additional terms in equations with respect to Einstein’s gravity, playing

the role of eﬀective pressures and energy densities, while preserving the energy

conditions for the matter content of the Universe. This naturally leads to the

question: what is the relationship between the cosmological bounce and po-

tential existence of wormholes? We address this question in the present paper

where we show that if the conditions for the cosmological bounce are established,

then wormholes can exist without any further modiﬁcation of ﬁeld equations or

without introducing any kind of exotic matter. To do this we ﬁrst present a

simple, very general and model-independent, description of bouncing cosmol-

ogy and then show how cosmological wormholes can be further constructed on

such spacetime. We then demonstrate that there is no violation of WEC in the

matter sector implied for such a kind of solutions.

The paper is organized in the following manner: in Section II we discuss how to

generally represent all possible types of cyclic cosmologies on FLRW spacetime

under the assumption of the stress-energy tensor conservation, in Section III we

discuss the wormhole geometry around the cosmological bounce, in Section IV

we derive ﬁeld equations for wormholes around the cosmological bounce, obtain

the solutions and discuss the WEC violation, and we ﬁnally conclude in V.

2 General approach to bouncing cosmology

In order to study the possibility of wormhole existence during the cosmological

bounce we ﬁrst need to present a general (i.e. model independent) description

of bouncing cosmological solutions. Here we follow the steps discussed in [39].

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CreationofwormholesduringthecosmologicalbouncePetarPavlović∗1andMarkoSossich†2,11InstituteforCosmologyandPhilosophyofNature,TrgsvetogFlorijana16,Križevci,Croatia2UniversityofZagreb,FacultyofElectricalEngineeringandComputing,DepartmentofPhysics,Unska3,10000Zagreb,CroatiaFebruary7,2025AbstractInthiswo...

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