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Cosmology with Eddington-inspired gravity
James H. C. Scargill*
Theoretical Physics, University of Oxford, Rudolf Peierls Centre, 1 Keble Road, Oxford OX1 3NP, United Kingdom
Maximo Banados†
Pontificia Universidad Catolica de Chile, Avenida Vicuna Mackema 4860, Santiago, Chile
Pedro G. Ferreira‡
Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom(Received 5 October 2012; published 27 November 2012)
We study the dynamics of homogeneous, isotropic universes which are governed by the Eddington-
inspired alternative theory of gravity which has a single extra parameter, �. Previous results showing
singularity-avoiding behavior for � > 0 are found to be upheld in the case of domination by a perfect fluid
with equation of state parameter w> 0. The range � 13 <w< 0 is found to lead to universes which
experience unbounded expansion rates while still at a finite density. In the case � < 0 the addition of
spatial curvature is shown to lead to the possibility of oscillation between two finite densities. Domination
by a scalar field with an exponential potential is found to also lead to singularity-avoiding behavior when
� > 0. Certain values of the parameters governing the potential lead to behavior in which the expansion
rate of the universe changes sign several times before transitioning to regular general relativity-like
behavior.
DOI: 10.1103/PhysRevD.86.103533 PACS numbers: 98.80.�k
I. INTRODUCTION
In 1924 Eddington proposed a theory of gravitybased solely on a connection field without introducinga metric. A particular extension of Eddignton’s theoryincluding matter fields was recently considered inRef. [1] (see also Ref. [2]). In a vacuum, this theoryreproduces ‘‘Eddington-inspired’’ (or just ‘‘Eddington’’)gravity and is completely equivalent to general relativity(GR), a fact which should be contrasted with othermodified gravity theories in which vacuum deviationsare expected (e.g., metric fðRÞ theories [3]). On theother hand deviations from GR are seen in regions ofhigh density. Where one would expect singularities suchas the interior of black holes or the birth of the uni-verse, in Eddington-inspired gravity often these areavoided [1,4,5] (although other singularities may appearin an astrophysical context [6]). The appearance of suchbehavior in a purely classical theory such as Eddingtongravity is interesting and exciting.
The cosmological implications of this theory were ini-tially considered in Ref. [1], and studied in more depth inRefs. [5,7]; it was found that universes containing ordinarymatter avoid singularites, with the exact behavior depend-ing on the sign of one of the parameters of the theory, andbeing either a regular bounce (from contraction to expan-sion) or ‘‘loitering’’ around a minimum size before tran-sitioning to GR-like expansion. It has also recently been
shown that tensor perturbations may render the regular
behavior of such cosmological models unstable near the
minimum scale (either in the asymptotic past or at the
bounce [8]).In this paper we confirm those findings [in the addition
of spatial curvature to the Friedman-Robertson-Walker
(FRW) metric], and present other interesting behavior
viz. universes which experience unbounded expansion
rates at finite density akin to undergoing a second big
bang/’’cosmic hiccup,’’ and closed universes which oscil-
late between two finite values of the density. Finally we
investigate the dynamics of universes in which the domi-
nant contribution to the stress-energy tensor is due to a
scalar field with an exponential potential, as has been done
in the case of GR [9].This paper is structured as follows: Sec. I A covers
some basic definitions and the governing field equations;Sec. I B exhibits the Friedman equation and qualitativelycompares it to GR; Sec. II examines solutions to thiswith domination by one kind of perfect fluid; Sec. II Bdemonstrates the ‘‘loitering’’ behavior which has previ-ously been observed; Sec. II C investigates the ‘‘cosmichiccup’’ mentioned above; Sec. II D exhibits the oscillat-ing universes; Sec. III covers the case of domination bya scalar field; Sec. III A investigates the ‘‘loitering’’behavior which has been observed for the case of regularmatter fields and which shows up again in the case ofscalar fields; Sec. III B exhibits some universes whichshow qualitatively new behavior—expansion followedby contraction followed by expansion. In Sec. IV weconclude.
*[email protected]†[email protected]‡[email protected]
PHYSICAL REVIEW D 86, 103533 (2012)
1550-7998=2012=86(10)=103533(12) 103533-1 � 2012 American Physical Society
A. Definitions and field equations
The action governing Eddington gravity is [1]
S½g;�;��¼ 2
�
Zd4x
� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijgþ�Rð�Þj
q��
ffiffiffiffiffiffijgj
q �þSM½g;��;
(1)
(throughout this paper we use reduced Planck units,c ¼ 8�G ¼ 1) where Rð�Þ denotes the symmetric part ofthe Ricci tensor, and � is a dimensionless constant whichmust satisfy � ¼ 1þ �� if GR is to be recovered in thelimit of �R � g (though the cosmological constant will beassumed negligible in the following). Gravitational theo-ries with such a Born-Infeld—like structure have previ-ously been investigated in Ref. [10]. Matter can, of course,be coupled to the connection as well as the metric, howeverfor simplicity here it is assumed to only couple to g. Twofield equations are derived from this action:
q�� ¼ g�� þ �R��ðqÞ; (2)
ffiffiffiffiffiffijqj
qq�� � �
ffiffiffiffiffiffijgj
qg�� ¼ ��
ffiffiffiffiffiffijgj
qT��; (3)
in which q is an auxiliary metric which is compatible withthe connection i.e., ��
��¼12q
��ð@�q��þ@�q���@�q��Þand q�� � ½q�1���.
The action (1), and its equations of motion (2) and (3)can also be derived from the following bigravity theory:
I½g; q� ¼Z ffiffiffi
qp ðq��R��ðqÞ � 2�Þ
þ�ð ffiffiffiq
pq��g�� � 2
ffiffiffig
p Þ þ ffiffiffig
pLmðgÞ (4)
with
� ¼ 1
�: (5)
Varying (4) with respect to q�� and g�� one obtains a set of
equations fully equivalent to (2) and (3). This form of theaction is particularly interesting because it shows a remark-able close relation with the recently discovered family ofunitary massive gravity theories [11,12]. Indeed, massivegravities are built as bigravity theories where the potentialis a linear combination of the symmetric polynomials ofthe eigenvalues of the matrix
� ¼ffiffiffiffiffiffiffiffiffiffiffiq�1g
q: (6)
For this class of potentials the theory has no Boulware-Deser instability and, in particular, the massive sectorcarries 5 degrees of freedom instead of 6. Besides thecosmological constants [which also appear in (4)], onesuch symmetric polynomial is
ffiffiffiq
p ðTrð�2Þ � ðTr�Þ2Þ ¼ ffiffiffiq
p ðq��g�� � ðTr�Þ2Þ: (7)
The first term is precisely the interaction between q��
and g�� appearing in (4), resulting in Eddington theory
coupled to matter. The second piece (more difficult tocompute because the square root has to be taken beforethe trace) is not part of the Eddington theory. We find itintriguing that Eddington’s theory coupled to matter is soclose to the recently discovered stable massive gravitytheories. We shall exploit this connection elsewhere.
B. Friedman equation
The FRW metric is
ds2 ¼ �dt2 þ aðtÞ2�
dr2
1� kr2þ r2ðd2 þ sin2d2Þ
�;
(8)
and we take the auxiliary metric to be
q��dx�dx�¼�UðtÞdt2þaðtÞ2VðtÞ
��
dr2
1�kqr2þr2ðd2þsin2d2Þ
�: (9)
The energy-momentum tensor is
T00¼Xi
�iþ� and Tij¼a2�X
i
PiþP
��ij; (10)
where fig indicate the regular matter and the scalar field.Finally,
� ¼ 1
2_2 þ VðÞ and P ¼ 1
2_2 � VðÞ: (11)
The Eddington gravity field equations enforce kq ¼ k
and
U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� �PTÞ31þ ��T
sand V ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ ��TÞð1� �PTÞ
q;
(12)
where �T ¼ Pi�i þ � þ� and PT ¼ P
iPi þ P ��.
The energy-momentum conservation equations are as inGR, so
€þ 3H _þ V0ðÞ ¼ 0; (13)
_�i þ 3H�ið1þ wiÞ ¼ 0: (14)
In the general case
H � _a
a¼ 1
F
0@�
ffiffiffiffiG
6
s� B
1A (15)
with
SCARGILL, BANADOS, AND FERREIRA PHYSICAL REVIEW D 86, 103533 (2012)
103533-2
G ¼ 1
�
�1þ 2U� 3
U
V� 6�k
a2
�; (16)
F ¼ 1��3�
�Xi
ð1� wi � �ðwi�T þ PTÞÞ�ið1þ wiÞ
� �ð�T þ PTÞ _2
���4ð1þ ��TÞð1� �PTÞ; (17)
B ¼ 1
2�
V 0 _1� �PT
: (18)
One key thing to note is that the scalar field does notsimply enter in the same way as other forms of matter (as isthe case in GR), which leads to the interesting result thatalthough there are two solutions for H, they are not simplyopposite signs, as in GR and Eddington gravity without ascalar field.
If one assumes that signð _VÞ ¼ �signðHÞ, i.e., as theuniverse increases in size, the potential energy of the scalarfield decreases, then in fact the values for H are simply ofdifferent signs. However while this assumption may holdfor the low-density GR-like stage of evolution, it fails inthe high density case with some interesting consequencesas laid out in Sec. III B
Another departure from GR is that spatial curvaturecannot be considered a fluid, since it does not enter inthe same way as a fluid with w ¼ � 1
3 (which it would have
to have in order to evolve as curvature).Finally, in the low-density and curvature limit (�i, �,
�, k=a2 � ��1) GR is recovered to leading order:
H ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�T
3� k
a2
sþ �
8<:�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�T
3� k
a2
s �3
4
Xi
�ið1� w2i Þ
þ 1
12
��T
3� k
a2
��1�15
8P2T � 9
4�TPT � 17
8�2T
þ 6k
a2ðPT þ �TÞ
��� 1
2V0 _
9=;þOðð��TÞ2Þ: (19)
II. DYNAMICS OFA UNIVERSE WITHOUTA SCALAR FIELD
We first consider the dynamics of universes which do notposses scalar fields, have a negligible cosmological con-stant and, for simplicity, only one type of perfect fluid. TheFriedman equation in this case is
H2 ¼ G
6F2(20)
¼ 8
3�
�ð1þ3wÞ ���2þ2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ ��Þð1�w ��Þ3
q�6 �k
a2ð1�w ��Þ
�
� ð1þ ��Þð1�w ��Þ2ð4þð1�wÞð1�3wÞ ��þ2wð1þ3wÞ ��2Þ2 ; (21)
where �� ¼ �� and �k ¼ �k.Note that setting w ¼ �1 (so that the universe contains
only a cosmological constant) reduces Eq. (21) to exactlythat which would be expected in GR, as it should be sinceEddington gravity exactly reproduces GR in a vacuum(with a cosmological constant).
A. Behavior of the Friedman equation
The square root in the numerator restricts the domain ofH2 and the result is shown in Table I.The requirement H2 � 0 fixes the sign of � and Table II
shows the results (for k ¼ 0), from which we see that theenergy density is positive in all but two regions of parame-ter space. This is an interesting departure from GR which,in the spatially flat case, always requires non-negativeenergy densities.As pointed out in Ref. [7], a final interesting quality of
(21) to note is that for certain ranges of w, H2 ! 1 forfinite �. Section II C investigates a universe which exhibitssuch behavior.Looking at the discriminant of the quadratic in the
denominator of (21), we see that the discontinuities do
not exist for 0:02 � 13 ð7� 4
ffiffiffi3
p Þ<w< 13 ð7þ 4
ffiffiffi3
p Þ �4:64. Outside of this region we can determine graphically,
TABLE I. Domain restrictions placed on H2 by the squareroot.
w Valid �� Range
w> 0 �1 �� 1w
w ¼ 0 �1 ���1 w< 0 �� 1
w or �1 ��w �1 �� �1 or 1
w ��
TABLE II. Sign of � in various regions of parameter space,with no spatial curvature. ��G satisfies Gð ��GÞ ¼ H2ð ��GÞ ¼ 0.(w ¼ �1 is special but also has � � 0.)
w �� Sign(�) Sign(�)
Any 0< �� þ þ�1<w �1< �� < 0 � þ�1<w< 0 ��G < �� < 1
w þ ��� < ��G � þ
w<�1 ��G < �� < 0 � þ1w < �� < ��G þ �
�� <�1 � þ
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given the constraints of Table I, in which regions thediscontinuities will appear (see Appendix A), and theresults are in Table III.
Points at which H2 ¼ 0 and dH2
d� � 0 will induce an
expanding universe to collapse (and vice versa) since theuniverse cannot evolve further on this trajectory (becauseH2 � 0) and the stationary point is unstable since onlyH2,and not the sign of H, is fixed by the physics. Such‘‘bounce’’ points exist in GR (if k > 0), and in Eddingtongravity as well, i.e., the points �� ¼ �1 and �� ¼ ��G. Weare not including � ¼ 0 here because, although H2
behaves as required, starting from a nonzero density thispoint cannot be reached in a finite time.
From (21) we see that at �� ¼ w�1, H2 ¼ 0 and dH2
d� ¼ 0.
This novel type of stationary point leads to the ‘‘loitering’’behavior remarked upon in Refs. [1,5,7] in which theuniverse asymptotically approaches (or recedes from) amaximum density/minimum size. From Table II we seethat for a positive energy density, this type of universe willonly exist for w> 0 and � > 0.
B. The case � > 0, w > 0
As seen in Fig. 1, this type of universe exhibits the‘‘loitering’’ behavior mentioned in the previous section.We also see that to a good approximation the evolution canbe split into two phases: the loitering phase in whichdeviations from GR are important, and the GR-like phasein which the dynamics of the expansion are approximatelyas given by GR.
1. Loitering phase
Close to the maximum density �� � ��B ¼ w�1 the terminvolving k is much smaller than the other terms in thenumerator of (21), and so we can neglect (small) spatialcurvature during the loitering phase. Expanding about themaximum density, or equivalently the minimum scalefactor aB, to highest order the Friedman equation anddeceleration parameter become
H2 ¼ 8
3�
�w� ��
3ð1þ wÞ�2 ¼ 8
3�
��a
aB
�2; (22)
qdec ¼�
w� ��
3ð1þ wÞ��1 ¼ �
��a
aB
��1; (23)
where � �� ¼ ��� ��B and �a ¼ a� aB. Using H ¼ _aa ¼
_�aaBþ�a , (22) can be solved to get
aðtÞaB
¼ 1þ exp
0@
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A; (24)
qdec ¼ � exp
0@�
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A: (25)
Thus we see that the universe undergoes (accelerating)exponential expansion away from the minimum scale fac-tor during this phase. The form of aðtÞ is reminiscent of thatobserved in GR under domination by a cosmological con-stant, specifically with � ¼ 8��1.
TABLE III. Regions in which discontinuities of H2 appear and are not outside of the allowed range of ��.
w<�1 w ¼ �1 �1<w<� 13 w ¼ � 1
3 � 13 <w< 0 0 w< 4:64 . . . 4:64::: w
��G < �� < 0 � � � �� < ��G � � � 0< �� � � � �1< �� < 0
0
0.03
0.06
0.09
0.12
0.15
0 0.2 0.4 0.6 0.8 1
H2
/ B
0
1
2
3
4
a/a B
Exact Soln.Loitering Approx.GR Approx.
0
0.4
0.8
1.2
0 5 10 15
t
UV
FIG. 1. Left: Expansion rate of the universe against density [normalized by the maximum density �B ¼ ð�wÞ�1] for � ¼ 1, w ¼1=3, k ¼ 0. Right top: Scale factor (normalized by the minimum scale factor aB) against time (in reduced Planck units) for the fullnumerical solution to the Friedman equation, and approximations in two phases: loitering, as in Sec. II B 1, and GR-like low-densitylimit. Right bottom: Behavior of the parameters describing the auxiliary metric (for the same universe).
SCARGILL, BANADOS, AND FERREIRA PHYSICAL REVIEW D 86, 103533 (2012)
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Calculating the particle horizon for such a universe wefind
rpðtÞ ¼ limt�!�1
Z t
t�
dt0
aðt0Þ ¼ limt�!�1
t� t�aB
; (26)
which diverges. Again this is similar to GR under domi-nation by a cosmological constant.
In Ref. [4] the authors note that Eddington gravity can becharacterized as making the gravitational force repulsive atshort distances (or high densities), which would agree withthe behavior which we see here. That is, the maximumdensity is reached as the repulsive effects of Eddingtongravity oppose the traditional attractive effects of the mat-ter density, whose repulsive effects take a form similar to acosmological constant with � ¼ 8��1.
2. Behavior of the auxiliary metric
From Eq. (9) we see that the auxiliary metric q�� is
identical to the metric g�� except that the temporal and
spatial components of q are modulated by the functions Uand V, respectively. Alternatively, by factoring out U, itcan be considered to be related to a FRW metric with
modified scale factor aq ¼ affiffiffiVU
q, by an overall conformal
transformation U. Thus U and V, or U and aq explain how
the space described by the auxiliary metric behaves.Figure 1 shows the behavior ofU and V. As expected, in
the GR regime U, V ! 1. On the other hand, close to themaximum density U, V ! 0, and the space described by qbecomes vanishingly small. Interestingly, V reaches amaximum value just before the GR regime begins.
Expanding close to the maximum density we find
U ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið3ð1þ wÞÞ3
2
sexp
0@32
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A; (27)
V ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð1þ wÞ
2
sexp
0@12
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A; (28)
aq ¼ aB3ð1þ wÞ exp
0@� 1
2
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A: (29)
And so we see that moving towards the maximumdensity, the temporal size decreases more rapidly thanthe spatial, with the result that aq actually goes infinite.
3. The case w ¼ 0
This case exhibits the same loitering behavior at earlytimes, but must be treated separately since the density isunbounded; expanding about a ¼ 0, to leading order wefind
H2 ¼ 8
3�; (30)
qdec ¼ �1; (31)
which is identical to GR with a cosmological constant� ¼ 8��1. Therefore the particle horizon will diverge asin the case w> 0.The behavior of U and aq is the same as in the w> 0
case, however in contrast, V diverges as t ! �1. SeeAppendix B 1 for the explicit functional forms.
C. The case � > 0, � 13 < w < 0
Firstly it should be noted that new physics (beyondEddington gravity) is required in order to have a fluidwith w< 0 dominating in the early universe; this problemis not considered here.As shown in Table III, in such a universe the expansion
rate diverges at a finite density given by the positive root ofthe denominator of Eq. (21)
�� ¼ ð1� wÞð1� 3wÞ þ ð1þ wÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 42wþ 9w2
p
4jwjð1þ 3wÞ :
(32)
Expanding around this point, to leading order we find
H2 ¼ 8
3�f2w
��a
a
��2(33)
with
f2w ¼�ð1þ 3wÞ �� � 2þ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ ��Þð1� w ��Þ3
q
� 6 �k
a2ð1� w ��Þ
�1
ð3 ��ð1þ wÞÞ2
� ð1þ ��Þð1� w ��Þ2ðð1� wÞð1� 3wÞ þ 4wð1þ 3wÞ ��Þ2
: (34)
This can be solved as in Sec. II B 1 to get
aðtÞa
¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2fw
ffiffiffiffiffiffi8
3�
sjt� tj
vuut: (35)
Thus at this critical point the universe in some senseundergoes a second big bang, with expansion similar tothat of GR with domination by radiation (a / ffiffi
tp
). Figure 2demonstrates this universe.Expanding Eq. (21) for high densities we find
H2 ¼ 4
3
jwj32ð1þ 3wÞ2 �: (36)
The expansion in the early history of this universe is also
GR-like, albeit with a modified density ~� ¼ 4jwj32ð1þ3wÞ2 �, and
the spatial curvature can be neglected. The scale factor as afunction of time is then
COSMOLOGY WITH EDDINGTON-INSPIRED GRAVITY PHYSICAL REVIEW D 86, 103533 (2012)
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aðtÞa
¼0@
ffiffiffiffiffiffi~�3
s3ð1þ wÞ
2t
1A
23ð1þwÞ
: (37)
In order to better understand the dynamics of the uni-verse during this second big bang/’’cosmic hiccup’’ weshould consider the horizon between two times t� < t <tþ close to t:
rðt�; tþÞ ¼ tþ � t�a
þOððt� � tÞ2Þ: (38)
Its well-behaved nature should not be surprising, however,since the nature of the expansion during the ‘‘hiccup’’ hasalready been noted.
1. Length of first life
An interesting quantity to consider is t, the timebetween the birth of the universe and the ‘‘hiccup.’’
Naively, since �� ! 1 for w ! 0, � 13 , it might be
expected that t ! 0 in these limits.We have
t ¼Z t
0dt ¼
Z a
0
da
aHðaÞ /Z 1
�
d�
�Hð�Þ : (39)
While this cannot be evaluated analytically in the generalcase, certain approximations can be made. For w ! � 1
3 ,
� / 11þ3w , and H is given by Eq. (36) and so the behavior
of Eq. (39) can be approximated by t /R�1 ð1þ3wÞd�
�3=2 ¼2ð1þ3wÞ�1=2
/ ð1þ 3wÞ3=2 ! 0, as expected.
However for w ! 0, � ! 12jwj and so Eq. (39) behaves
t /R�1 d�
jwj3=4�3=2 ¼ 2
jwj3=4�1=2
� jwj�1=4 ! 1, which defies
the naive expectations. This occurs because the expansionrate of the universe heads towards zero at high densitiesmore quickly than �� ! 1, as indeed it should if the high
0
20
40
60
80
100
0 2 4 6 8 10
H2
/ *
0
0.5
1
1.5
2
a/a *
Exact Soln.Hiccup Approx.
0 2
4
6
8
0.0 0.1 0.2 0.3 0.4 0.5t
UV
FIG. 2. Left: Expansion rate of the universe against density (normalized by the critical density �) for � ¼ 1, w ¼ �1=6, k ¼ 0.Right top: Scale factor (normalized by the critical scale factor a) against time (in reduced Planck units) for the full numerical solutionto the Friedman equation, and an approximation close to the critical density. Right bottom: Behavior of the parameters describing theauxiliary metric (for the same universe).
0
0.02
0.04
0.06
0.08
0.0 0.2 0.4 0.6 0.8 1.0
⏐⏐ H
2
/ B
1
1.5
a/a B
Exact Soln.High Density Approx.GR Approx.
0
0.5
1
1.5
0 5 10 15
t
UV
FIG. 3. Left: Expansion rate of the universe against density (normalized by the maximum density �B ¼ ���1) for � ¼ �1, w ¼1=3, k=a2B ¼ 1=10. Right top: Scale factor (normalized by the minimum scale factor aB) against time (in reduced Planck units) forthe full numerical solution to the Friedman equation, and approximations in two phases: close to the maximum density, and GR-likelow-density limit. Right bottom: Behavior of the parameters describing the auxiliary metric (for the same universe).
SCARGILL, BANADOS, AND FERREIRA PHYSICAL REVIEW D 86, 103533 (2012)
103533-6
density behavior observed in Sec. II B 3 is to be recoveredwhen w ¼ 0.
2. Behavior of the auxiliary metric
In the high-density limit the overall size of the spacedescribed by q diverges, but the combined effect of U andV is that aq remains proportional to the regular scale factor.
At the ‘‘hiccup’’ U and V as functions of �� or a aresmooth and well behaved, but as functions of t they ofcourse exhibit a ‘‘kink’’ just as aðtÞ does (see Fig. 2).
Explicit functional forms in these two limits can befound in Appendix B 2.
D. Oscillating universes
In Refs. [1,5,7] it is noted that for � < 0, w> 0 theuniverse undergoes a ‘‘regular’’ bounce (as describedbelow) in which, in a finite time, a collapsing universereaches a maximum density and then expands. In GR,universes with positive spatial curvature can undergo abounce in the opposite direction.
The addition of spatial curvature to Eddington cosmol-ogy thus leads to the interesting prospect of oscillatorybehavior. Figure 3 demonstrates such an oscillatoryuniverse.
1. Amplitude of the oscillations
The maximum density �B ¼ ���1 is independent of wand k, and is caused by the (1þ ��) factor in the numeratorof Eq. (21). The minimum density, on the other hand, doesdepend on w and k (and �), and is the solution to G ¼ 0,which explicitly is
ð1þ 3wÞ ��� 2þ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ ��Þð1� w ��Þ3
q� 6 �ka�2ð1� w ��Þ ¼ 0: (40)
Unfortunately this cannot be solved analytically in thegeneral case, however if we assume that the universe hasexpanded sufficiently before expansion halts then we cansolve this in the GR regime to get
amax
aB¼
�3k
a2B
�� 11þ3w
;�min
�B
¼�3k
a2B
�3ð1þwÞ1þ3w
: (41)
From Fig. 4 we see that this is indeed a good approxi-mation for k=a2B & 0:2.
Close to the maximum density we have
H2 ¼ 8
3j�j�1þ 2 �k
a2B
��a
aB; (42)
which is solved to give
aðtÞaB
¼ 1þ 2
3j�j�1þ 2 �k
a2B
�jt� tBj2: (43)
Note that this is independent of the type of matter whichfills the universe (just as in Sec. II B 1), and so we under-stand the expansion is being driven by �.Bounds can be placed on k by examining the Friedman
equation in these two limits [since the behavior of Eq. (21)as a function of �� and k is smooth] and requiring H2 � 0.The GR-like regime gives a lower bound on k whileEq. (42) gives an upper bound, which combine to give
0<k
a2B<
1
2j�j : (44)
2. Period of the oscillations
In principle the period of the oscillations can be calcu-lated from
Tosc ¼ 2Z Tosc=2
0dt ¼ 2
Z amax
aB
da
aHðaÞ : (45)
In practice this integral cannot be performed analytically(or good enough analytical approximations made) and so itmust be evaluated numerically, as shown in Fig. 4.We see that Toscðw ¼ 1
3Þ< Toscðw ¼ 0Þ, which makes
sense since the amplitude of the oscillations for the formercase is always smaller than in the latter. Also note thatTosc ! 0 for k=a2B ! 1=2. From Eq. (42) we see that theexpansion rate goes to zero at this point, and so fromSec. II C 1 we might expect that Tosc ! 1, however inthis case the extra size by which the universe must expandalso heads to zero (and clearly does so more quickly thanthe expansion rate).
min
/B
w = 0 Exact Soln.w = 0 GR Approx.w = 1/3 Exact Soln.w = 1/3 GR Approx.
0.0 0.1 0.2 0.3 0.4 0.5
Tos
c
k/a2B
w = 0w = 1/3
0
0.2
0.4
0.6
0.8
1
0 5
10
15
20
FIG. 4. Top: Amplitude (minimum density, normalized bymaximum density) of the oscillations described in Sec. II Dagainst spatial curvature of the universe for � ¼ �1 and variousvalues for the equation of state parameter. Full numerical solu-tions, as well as approximations in the GR regime from Eq. (41).Bottom: Period of the oscillations (in reduced Planck units) forthe same universes.
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3. Behavior of the auxiliary metric
Close to the maximum density the spatial size de-creases to zero as the universe reaches is minimum size,while the temporal component of the auxiliary metricdiverges. See Appendix B 3 for approximate functionalforms for these.
III. DYNAMICS OF A UNIVERSE WITH ASCALAR FIELD
We now consider the cosmology when a scalar field isintroduced; the ordinary matter content, the cosmologicalconstant and the spatial curvature are considered negli-gible. Written in terms of and VðÞ the FRWequation is
H ¼24�
�1� �
2_2 þ �V
� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ �
2_2 þ �V
��1
3
�_2 � V � 1
�
�1þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ �
2_2 þ �V
��1� �
2_2 þ �V
�3
s ���vuut
� �
2
�1þ �
2_2 þ �V
�V 0 _
35 1
1þ 2�V þ �2ð12 _4 þ V2Þ : (46)
The positive sign is taken, as this gives a universe which isexpanding in the GR regime. The conservation equationgoverning the scalar field is
€þ 3H _þ V 0ðÞ ¼ 0: (47)
We restrict ourselves to an exponential potential for thescalar field
VðÞ ¼ V0 expð��Þ (48)
with V0, � > 0. Unless explicitly included, the parameterdescribing deviations from GR (�) is set to 1.
A. Loitering behavior
For � > 0 numerical solutions of Eq. (46) show quali-tatively similar behavior to the case without scalar modes,viz. ‘‘loitering’’ of the scale factor around a minimumvalue followed by a transition to GR-like power-law ex-pansion. See Fig. 5, top left panel.The solution for ðtÞ also shows two regimes with a
transition between them (Fig. 5, bottom right panel). TheGR regime shows the expected logarithmic time depen-dence [9], while the Eddington/loitering regime shows / �t.
0
1
2
3
4
a
V0=1/4, =2V0=1/4, =3V0=1/2, =2V0=1/2, =3
0
0.4
0.8
1.2
1.6
-3 -2 -1 0 1 2 3 4 5
t
0
0.2
0.4
0.6
0.8
1
-0.8
-0.4
0
0.4
0.8
-3 -2 -1 0 1 2 3 4 5
w
t
FIG. 5. Scale factor (top left), scalar field (bottom left), energy density (top right) and equation of state parameter (w � P=�)(bottom right) as a function of time (in reduced Planck units) for various values of the free parameters.
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1. Initial conditions and free parameters
It is appropriate first to briefly discuss the initial con-ditions used to produce the solutions plotted above as wellas the effect on these that is had by altering the freeparameters of the system.
As can be seen from Fig. 5, the chosen conditions are
aðt ¼ 0Þ ¼ 1; ð0Þ ¼ 0; _ð0Þ ¼ 0: (49)
The first is uncontroversial since the scale factor canalways be rescaled by a constant. The absolute value ofthe scalar field only appears in the form for its potentialenergy, and thus adjusting the field by a constant, which isthe action of the second initial conditions (i.c.), only hasthe effect of changing V0. The theory is obviously invariantunder a constant time shift and the third i.c. just fixes this.
The reader may be worried that using _ ¼ 0 as an i.c.eliminates from consideration any solutions which showmonotonically increasing field values, however such solu-tions would not exhibit the maximum density and loiteringbehavior (since if is unbounded from below the densitywill also be unbounded, and if tends to some constantvalue, the pressure would be negative and hence not satisfyP ¼ ��1).
All that is left is to describe the potential, whichleaves two free parameters: V0 and �. From the figuresthus far presented we can roughly evaluate the effect ofmaking the potential steeper (increasing �) or deeper(increasing V0).
The late time (GR-like) behavior is primarily governedby the value of �, whereas the early time behavior isinfluenced by V0. The reason for the latter is that the energydensity of the field is already fixed as t ! �1, and pickinga value for V0 fixes it at t ¼ 0 (or wherever the transi-tionary period is located), and thus the behavior in betweenthese two points will be greatly influenced by the size ofthe change in the energy density.
Furthermore one can roughly say that steeper and deeperpotentials lead to quicker transitions from the loitering tothe GR phase.
2. Loitering regime
Let us investigate this loitering behavior in more detail.As in the case with ordinary matter the loitering regimecorresponds to P ! ��1 (as t ! �1); we also see that ! 1 and hence VðÞ ! 0. Thus one has � � P and
! �ffiffiffi2�
qtþ const.
Evaluating the Friedman equation in this regime, onefinds
HðP ¼ ��1Þ ¼ 2ð1þ ��ÞV 0
3� _3¼ 1
3
ffiffiffiffiffiffi2�
pV 0: (50)
Therefore the equation of motion for the scalar field
reduces to €þ 3V 0 ¼ 0. The solution (setting � ¼ 1) is
ðtÞ ¼ � ffiffiffi2
p ðt� t0Þ þ 1
�½2 lnð12V0 þ e
ffiffi2
p�ðt�t0ÞÞ � 4 ln2�;
(51)
aðtÞ � amin exp
�16V0
3exp
ffiffiffi2
p�ðt� t0Þ
�: (52)
The dynamic nature of the scalar field adjusts the loiter-ing behavior of the scale factor from the simple exponen-tial behavior seen in the case of regular matter to the moreextreme double exponential.
3. Transitionary regime
Whereas in the case of ordinary matter fields the evolu-tion is well described without having to denote a separatetransitionary regime (between the high-density loiteringbehavior and the low-density GR-like regime), that is notthe case here.The transition from linearly decreasing field values to
logarithmically increasing field values is naturally centered
on _ ¼ 0. Hence during the transition P � �� the scalarfield has the equation of state of a cosmological constant.This is readily observed in plots of �ðtÞwhich show that thedensity is stationary around the transition. See Fig. 5, topright panel.The bottom right panel of this figure also confirms this,
as well as the above comment that P � � in the loiteringregime (w ! 1 as t ! �1, regardless of the values of thefree parameters).
4. Density scaling
Finally it is interesting to look at the plot of ln� againstlna (Fig. 6). This shows that during the loitering phase thedensity scales as � / a�6, which agrees with the fact thatw � 1; the scalar field is, to use the terminology used inRef. [9], undergoing kination (kinetic energy domination).This is followed by the expected cosmological constantliketransitionary phase in which the density is stationary. Thefield energy density shows the expected scaling in the GR
phase (� / a��2if � <
ffiffiffi6
pand � / a�6 otherwise).
Figure 1 from Ref. [9] shows the evolution of the energydensity of a scalar field in a GR universe populated by theaforementioned scalar field (with � ¼ 4) as well as radia-tion and cold matter. The scalar field is initially dominantand scales accordingly (� / a�6), then it becomes subdo-minant and scales like a constant before finally tracking(i.e., scaling as) the cold matter (� / a�3) which is thendominant.This bears a remarkable resemblance to Fig. 6, except
for two major differences:
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(1) The behavior in the Eddington case is reversed. The‘‘natural’’ scaling (i.e., that which one would expectin GR based on the value of �) is preceded by aperiod in which the scaling is fixed (by somethingother than �).
(2) In the Eddington case such behavior is achievedwithout another fluid for the field to track.
B. Expand-contract-expand behavior
As noted in Sec. I B, the two solutions for the Hubbleparameter are not in general simply of the same magnitudebut with opposite sign. This leads to the possibility thatthere are universes which at some point are stationary,and then instead of contracting and retracing their evolu-tionary history in reverse, they contract but trace a newhistory [13].
In fact since the solutions being investigated are expand-ing both in their early and late histories, they wouldundergo expansion followed by contraction followed byexpansion.
From Fig. 7 we see that such behavior can be achievedboth by making the potential steeper and by making itdeeper. The latter can easily be understood since V0, whichcontrols the depth of the potential, also controls the energydensity of the scalar field at its transitionary point. Thuschoosing a sufficiently large V0 will ensure that the densityat this point is larger than the density in the distant past(loitering phase), and hence the universe must undergocontraction (see Fig. 7, bottom panel).
That steeper potentials induce this behavior canbe understood mathematically from the fact that the ‘‘B’’term in the Friedman equation [Eqs. (15) and (18)] isproportional to V0, and hence to �. Physically, perhapsone can imagine the cause to be that a potential whichvaries quickly enough with leads the scalar field to‘‘overshoot’’ the value which would ensure energy densitymonotonically decreases with time; this larger densitywould then be associated with a contraction.
It is interesting to note that plots of the field value, or ofthe equation of state parameter, do not obviously show anyeffect of the period of contraction or differ qualitativelyfrom the corresponding plots for free parameter valueswhich show the ordinary behavior (Fig. 5).Unfortunately we have yet to determine the critical
values of V0 and � which guard the transition into suchbehavior.
IV. CONCLUSION
In this paper we have shown that the singularity-avoiding ‘‘loitering’’ behavior observed for � > 0 anddomination by a perfect fluid is upheld for w> 0 andthat this result is not affected by the presence of spatialcurvature. This behavior shows remarkable similarities tothe behavior in GR with a cosmological constant � ¼8��1. Similar behavior is also seen in the case of domina-tion by a scalar field with an exponential potential. Thescaling of the density with the scale factor in such uni-verses is reminiscent of that observed in a GR universewith such a scalar field and (initially more slowly scaling)ordinary matter. The similarities to GR in each of thesecases deserve further investigation which will shed light onthe high density behavior of Eddington gravity.
-0.5 0 0.5 1 1. 5
log
log a
V0=1/4, =2V0=1/4, =3V0=1/2, =2V0=1/2, =3
-3
-2.5
-2
-1.5
-1
-0.5
0
FIG. 6. Energy density as a function of scale factor for variousvalues of the free parameters.
H
V0=3/4, =4V0=2, =2V0=2, =4
at
-1
-0.5
0
0.5
1
1.5
0
1
2
0
0.5
1
1.5
2
-2 -1 0 1 2
FIG. 7. Hubble expansion parameter (top), scale factor(middle) and energy density (bottom) as a function of time (inreduced Planck units) for various values of the free parameters,showing universes which expand-contract-expand.
SCARGILL, BANADOS, AND FERREIRA PHYSICAL REVIEW D 86, 103533 (2012)
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The (past) singularity avoiding behavior of this theoryshould be compared to that examined in Ref. [14] in whichhigher derivative terms in fðRÞ theories lead to future finitetime singularity avoidance. The nonlinear structure of theEddington gravity action introduces such terms and so wecan perhaps qualitatively understand that this leads tosingularity avoidance in a similar manner.
Finally as was noted in Sec. I B, in the low-density limit(� � ��1, with ��1 * 1018 kgm�3 [15]) GR is recoveredand so the theory is indistinguishable from �CDM (orsimilar dark energy models) in this cosmological arena.
For both perfect fluids and scalar fields we have alsodemonstrated novel behavior unlike that seen in GR. In theformer case we have the ‘‘cosmic hiccup’’ which isobserved for � > 0,� 1
3 <w< 0, as well as the oscillatory
universes for � < 0, w> 0, and positive spatial curvature.In the latter case we have shown that especially deep orsteep scalar field potentials lead the size of the universeoscillating while transitioning between the loitering andGR-like phases.
There appears no initial reason why the singularityavoiding behavior observed for � < 0 cannot also existfor a scalar field dominated universe. As explained above,symmetry around the bounce would be permitted in this
case ( _ ¼ 0 at the bounce). Furthermore the interplay
between the scalar field and regular matter fields couldlead to interesting behavior of the scalar field, as is the casein GR [9] and remains to be studied in detail.
ACKNOWLEDGMENTS
This research is supported by STFC, Oxford MartinSchool and BIPAC. M.B. was partially supported byFondecyt (Chile) Grants No. 1100282 and No. 1090753.
APPENDIX A: REGIONS IN WHICHDISCONTINUITIES APPEAR
Figure 8 shows, for a universe dominated by a singleperfect fluid, the locations in ð ��;wÞ parameter space ofdiscontinuities inH2. There are no discontinuities at all for
0:02 � 13 ð7� 4
ffiffiffi3
p Þ<w< 13 ð7þ 4
ffiffiffi3
p Þ � 4:64, and out-
side of this range they are given by
�� ¼ � ð1� wÞð1� 3wÞ � ð1þ wÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 42wþ 9w2
p
4wð1þ 3wÞ :
(A1)
Also shown are the regions which are forbidden due tothe square root in G (and the requirement that H2 is real).Combining these data we arrive at Table III.
w
Forbidden RegionsAsymptotesDiscontinuities
-2.5
-2
-1.5
-1
-0.5
0
0.5
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
11.5
12
12.5
13
13.5
14
14.5
15
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2
w
-2.5
-2
-1.5
-1
-0.5
0
0.5
4 4.5 5 5.5 6
FIG. 8. Hatched regions are those forbidden due to the square root in G; solid lines indicate the solutions to F ¼ 0 and so representthe locations of discontinuities in H2; dashed lines indicate the asymptotes to which the solid lines tend.
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APPENDIX B: APPROXIMATE FUNCTIONALFORMS FOR THE BEHAVIOR OF THE
AUXILIARY METRIC
1. The case � > 0, w ¼ 0
Defining ��ða ¼ a0Þ ¼ ��0, during the loitering phase wefind
U ¼ 1ffiffiffiffiffiffi��0
p��a
a0
�32 ¼ 1ffiffiffiffiffiffi
��0
p exp
0@32
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A; (B1)
V ¼ ffiffiffiffiffiffi��0
p ��a
a0
��32 ¼ ffiffiffiffiffiffi
��0
pexp
0@� 3
2
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A; (B2)
aq¼a0ffiffiffiffiffiffi��0
p ��a
a0
��32¼a0
ffiffiffiffiffiffi��0
pexp
0@�3
2
ffiffiffiffiffiffi8
3�
sðt� t0Þ
1A: (B3)
2. The case � > 0, � 13 < w < 0
In the high-density limit, for the early history of thisuniverse, we have
U ¼ jwj32 ��; V ¼ jwj12 ��; aq ¼ ajwj�12: (B4)
Where a is given by (37) and thus �� ¼ �� ffiffiffiffi
~�3
q3ð1þwÞ
2 t�2
.
The functional forms close to the critical point are
U ¼ U�1� 3 ��ð1þ wÞ
2
�1
1þ ��þ 3w
1� w ��
��a
a
�;
(B5)
V ¼ V�1� 3 ��ð1þ wÞ
2
�1
1þ ��� w
1� w ��
��a
a
�;
(B6)
aq¼aq�1þ
�1�3 ��ð1þwÞ
2
�1
1þ ��þ w
1�w ��
���a
a
�;
(B7)
and �aa
can be read off from (35).
3. The case � < 0, w > 0
Close to the maximum density we have
U ¼ ð1þ wÞffiffiffiffiffiffiffiffiffiaB3�a
r¼ ð1þ wÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
j�j�1þ 2 �k
a2B
�s �1
jt� tBj�1;
(B8)
V ¼ ð1þ wÞffiffiffiffiffiffiffiffiffi3�a
aB
s¼ ð1þ wÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2
j�j�1þ 2 �k
a2B
�sjt� tBj;
(B9)
aq ¼ 3�a ¼ 2aBj�j
�1þ 2 �k
a2B
�jt� tBj2:
(B10)
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