How to distinguish dark energy and modified gravity?

How to distinguish dark energy and modified gravity?

Hao Wei *

Department of Physics and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China

Shuang Nan Zhang

Department of Physics and Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China;Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China;

Physics Department, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA(Received 24 March 2008; published 11 July 2008)

The current accelerated expansion of our universe could be due to an unknown energy component (dark

energy) or a modification of general relativity (modified gravity). In the literature it has been proposed that

combining the probes of the cosmic expansion history and growth history can distinguish between dark

energy and modified gravity. In this work, without invoking nontrivial dark energy clustering, we show

that the possible interaction between dark energy and dark matter could make the interacting dark model

and the modified gravity model indistinguishable. An explicit example is also given. Therefore, it is

required to seek some complementary probes beyond the ones of cosmic expansion history and growth

history.

DOI: 10.1103/PhysRevD.78.023011 PACS numbers: 95.36.+x, 04.50.�h, 98.80.�k

I. INTRODUCTION

The current accelerated expansion of our universe [1–10] has been one of the most active fields in moderncosmology. There are very strong model-independent evi-dences [11] (see also e.g. [12]) for the accelerated expan-sion. Many cosmological models have been proposed tointerpret this mysterious phenomenon, see e.g. [1] for arecent review.

In the flood of various cosmological models, one of themost important tasks is to distinguish between them. Thecurrent accelerated expansion of our universe could be dueto an unknown energy component (dark energy) or amodification to general relativity (modified gravity)[1,13]. Recently, some efforts have focused on differenti-ating dark energy and modified gravity with the growthfunction �ðzÞ � ��m=�m of the linear matter density con-trast as a function of redshift z. Up until now, most cos-mological observations merely probe the expansion historyof our universe [1–10]. As is well known, it is very easy tobuild models which share the same cosmic expansionhistory by means of reconstruction between models.Therefore, to distinguish various models, some indepen-dent and complementary probes are required. Recently, itwas argued that the measurement of growth function �ðzÞmight be competent, see e.g. [13–29]. If the dark energymodel and modified gravity model share the same cosmicexpansion history, they might have different growth histor-ies. Thus, they might be distinguished from each other.

However, the approach mentioned above has been chal-lenged by some authors. For instance, in [30], Kunz andSapone explicitly demonstrated that the dark energy mod-

els with nonvanishing anisotropic stress cannot be distin-guished from modified gravity models [e.g. DGP (Dvali-Gabadadze-Porrati) model] using growth function. In [31],Bertschinger and Zukin found that if dark energy is gen-eralized to include both entropy and shear stress perturba-tions, and the dynamics of dark energy is unknown apriori, then modified gravity [e.g. fðRÞ theories] cannotin general be distinguished from dark energy using cos-mological linear perturbations.Here, we investigate this issue in another way. Instead of

invoking nontrivial dark energy clustering (e.g. nonvanish-ing anisotropic stress), it is of interest to see whether theinteraction between dark energy and cold dark matter canmake dark energy and modified gravity indistinguishable.In fact, since the nature of both dark energy and darkmatter are still unknown, there is no physical argumentto exclude the possible interaction between them. On thecontrary, some observational evidences of this interactionhave been found recently. For example, in a series ofpapers by Bertolami et al. [32], they show that the AbellCluster A586 exhibits evidence of the interaction betweendark energy and dark matter, and they argue that thisinteraction might imply a violation of the equivalenceprinciple. On the other hand, in [33], Abdalla et al. foundthe signature of interaction between dark energy and darkmatter by using optical, x-ray and weak lensing data from33 relaxed galaxy clusters. In [34], Ichiki and Keum dis-cussed the cosmological signatures of interaction betweendark energy and massive neutrino (which is also a candi-date for dark matter) by using cosmic microwave back-ground power spectra and matter power spectrum.Therefore, it is reasonable to consider the interaction be-tween dark energy and dark matter in cosmology. Sincedark energy can decay into cold dark matter (and vice*[email protected]

PHYSICAL REVIEW D 78, 023011 (2008)

1550-7998=2008=78(2)=023011(7) 023011-1 � 2008 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.78.023011

versa) through interaction, both expansion history andgrowth history can be affected simultaneously, similar tothe case of modified gravity. Thus, it is natural to askwhether the combined probes of expansion history andgrowth history can distinguish between interacting darkenergy models and modified gravity models. The answermight be no. So, it is required to seek some complementaryprobes beyond the ones of cosmic expansion history andgrowth history.

In this paper, we propose a general approach in Sec. II tobuild an interacting quintessence model which shares boththe same expansion history and growth history with themodified gravity model. In Sec. III, following this pre-scription, as an example, we explicitly demonstrate theinteracting quintessence model which is indistinguishablewith the DGP model in this sense. A brief discussion isgiven in Sec. IV, in which some suggestions to break thisdegeneracy are also discussed.

II. GENERAL FORMALISM

Quintessence [35,36] is a well-known dark energy can-didate. In this section, we consider the interacting quintes-sence model [37–42] and show how it can share both thesame expansion history and growth history with modifiedgravity models, without invoking nontrivial dark energyclustering.

We consider a flat Friedmann-Robertson-Walker uni-verse. As is well known, the quintessence is described bya canonical scalar field with a Lagrangian density L� ¼12 ð@��Þ2 � Vð�Þ. Assuming the scalar field � is homoge-

neous, one obtains the pressure and energy density forquintessence

p� ¼ 12_�2 � Vð�Þ; �� ¼ 1

2_�2 þ Vð�Þ; (1)

where a dot denotes the derivative with respect to cosmictime t. The Friedmann equation reads

3H2 ¼ �2ð�m þ ��Þ; (2)

where H � _a=a is the Hubble parameter; a ¼ ð1þ zÞ�1 isthe scale factor (we have set a0 ¼ 1; the subscript ‘‘0’’indicates the present value of corresponding quantity; z isthe redshift); �m is the energy density of cold dark matter(we assume the baryon component to be negligible); �2 �8�G. We assume that quintessence and cold dark matterinteract through [37–39]

_�m þ 3H�m ¼ ��Q�m_�; (3)

_�� þ 3Hð�� þ p�Þ ¼ �Q�m_�; (4)

which preserves the total energy conservation equation_�tot þ 3Hð�tot þ ptotÞ ¼ 0. The dimensionless couplingcoefficient Q ¼ Qð�Þ is an arbitrary function of �.Equation (4) is equivalent to

€�þ 3H _�þ dV

d�¼ �Q�m: (5)

Using Eqs. (2)–(4), one can obtain the Raychaudhuri equa-tion

_H ¼ ��2

2ð�m þ �� þ p�Þ ¼ ��2

2ð�m þ _�2Þ: (6)

It is worth noting that due to the nonvanishing interaction,�m does not scale as a�3. The above equations are asso-ciated with the expansion history. On the side of growthhistory, as shown in [38], the perturbation equation in thesubhorizon regime is

�00 þ�2þH0

H� �Q�0

��0 ¼ 3

2ð1þ 2Q2Þ�m�; (7)

where � � ��m=�m is the linear matter density contrast;�m � �2�m=ð3H2Þ is the fractional energy density of colddark matter; and a prime denotes a derivative with respectto lna. Note that in [38] the absence of anisotropic stresshas been assumed, namely, in longitudinal (conformalNewtonian) gauge the metric perturbations � ¼ �.Obviously, when Q ¼ 0, Eq. (7) reduces to the standardform in general relativity [14–18,24,43]

€�þ 2H _� ¼ 4�G�m�: (8)

In fact, Eq. (7) from [38] is valid for any Q ¼ Qð�Þ andgeneralizes the one of [39] which is only valid for constantQ. On the other hand, in modified gravity, the perturbationequation (8) has been modified to [18,21,22,43,44]

€~�þ 2 ~H _~� ¼ 4�Geff�m~�; (9)

where the quantities in modified gravity are labeled by atilde ‘‘ �’’; Geff is the effective local gravitational ‘‘con-stant’’ measured by Cavendish-type experiment, which istime-dependent. In general, Geff can be written as

Geff ¼ G

�1þ 1

3�

�; (10)

where � is determined once we specify the modifiedgravity theory. Equation (9) can be rewritten as

~� 00 þ�2þ ~H0

~H

�~�0 ¼ 3

2

�1þ 1

3�

�~�m

~�: (11)

Now, we require that the interacting quintessence modelshares both the same expansion history and growth historywith the modified gravity model. That is, we identify

H ¼ ~H and � ¼ ~�: (12)

Comparing Eq. (7) with Eq. (11), we find that

�Q�0�0 ¼ 3

2�

��1þ 1

3�

�~�m � ð1þ 2Q2Þ�m

�: (13)

Note that �m � ~�m in general. From Eq. (6), we have

HAO WEI AND SHUANG NAN ZHANG PHYSICAL REVIEW D 78, 023011 (2008)

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ð��0Þ2 ¼ �3�m � 2H0

H: (14)

We can recast Eq. (3) to

�0m ¼ �

�3þ 2

H0

Hþ �Q�0

��m: (15)

It turns out

�Q�0 ¼ �3� 2H0

H��0

m

�m

: (16)

From Eqs. (14) and (16), we obtain

Q2 ¼ ð�Q�0Þ2ð��0Þ2 ¼ ð3þ 2 H0

H þ �0m

�mÞ2

�3�m � 2 H0H

: (17)

Noting that � ¼ ~�, we can find � in Eq. (13) from Eq. (11).

Once ~�m, �, ~H, and corresponding ~� in the modifiedgravity are given, substituting Eqs. (16) and (17) intoEq. (13) and noting Eq. (12), we obtain a differentialequation for �m with respect to lna. After we find�mðlnaÞ from this differential equation, ��0ðlnaÞ can beobtained from Eq. (14), while H ¼ ~H. Then, by usingEqs. (14) and (16), QðlnaÞ ¼ ð�Q�0Þ=ð��0Þ is in hand.From Eq. (2), �� 2��=ð3H2Þ ¼ 1��m. Noting

Eq. (1) and using Eq. (14), we find the dimensionlesspotential of quintessence

U � �2V

H20

¼ 3E2

�1��m

2þ 1

3

H0

H

�; (18)

where E � H=H0. Notice that H0=H ¼ E0=E. By integrat-

ing ��0ðlnaÞ, we find �� as a function of lna. Therefore,we can finally obtain Q, U as functions of ��.

In fact, this general approach proposed here is just anormal reconstruction method. For any given modifiedgravity model, we can always construct an interactingquintessence model in the framework of general relativitywhich shares both the same expansion history and growthhistory with this given modified gravity model. As is wellknown, an interacting quintessence model can be describedcompletely by its potential Vð�Þ and the coupling Qð�Þ.For a given modified gravity model, its ~�m, �, and ~H are

known, whereas its corresponding ~� can be obtained fromEq. (11). Following the procedure described in the textbelow Eq. (17), one can easily reconstruct the dimension-less potential Uð��Þ [which is equivalent to the potentialVð�Þ obviously] and the coupling Qð��Þ [which is Qð�Þin fact] for the interacting quintessence model. Up untilnow, the desired interacting quintessence model whichshares both the same expansion history and growth historywith the given modified gravity model has been con-structed. Therefore, the cosmological observations mightbe unable to distinguish between them, unless other com-plementary probes beyond the ones of cosmic expansionhistory and growth history are used.

III. EXPLICIT EXAMPLE

In this section, we give an explicit example following theprescription proposed in Sec. II. Here, we consider theDGP braneworld model [45] (see also e.g. [18,21,22,46]),which is the simplest modified gravity model. Assumingthe flatness of our universe, in the DGP model (here weonly consider the self-accelerating branch), ~E � ~H= ~H0 isgiven by [18,21,22]

~E ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~�m0ð1þ zÞ3 þ ~�rc

qþ

ffiffiffiffiffiffiffiffi~�rc

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~�m0e

�3 lna þ ~�rc

qþ

ffiffiffiffiffiffiffiffi~�rc

q; (19)

where ~�rc is constant.~Eðz ¼ 0Þ ¼ 1 requires

~� m0 ¼ 1� 2ffiffiffiffiffiffiffiffi~�rc

q: (20)

Therefore, the DGP model has only one independentmodel parameter. The fractional energy density of matterin the DGP model reads

~� m ¼~�m0ð1þ zÞ3

~E2ðzÞ ¼~�m0e

�3 lna

~E2ðlnaÞ : (21)

In addition, the � in Eq. (10) for the flat DGP model isgiven by [18,21,22]

� ¼ � 1þ ~�2m

1� ~�2m

: (22)

Fitting the DGPmodel to the 192 SNIa data compiled byDavis et al. [10] which are joint data from ESSENCE [9]

and Gold07 [3], we find that the best-fit parameter ~�rc ¼0:170 with �2

min ¼ 196:128 while the corresponding ~�m0

is given by Eq. (20). Substituting this ~�m0 into Eqs. (19),

(21), and (22), ~�m, �, and ~E as functions of lna are known(since our main aim is to demonstrate the prescriptionproposed in Sec. II, we do not consider the errors, even

one can use any ~�m0 here for demonstration). Noting thatH0=H ¼ E0=E, following the prescription proposed inSec. II, we can easily construct the desired interactingquintessence model which shares both the same expansionhistory and growth history with the corresponding DGP

model. At the first step, we obtain � ¼ ~� from Eq. (11). As

is well known, ~�0 ¼ ~� ¼ a at z � 1 (see e.g. [14,15]).

Thus, we use the initial condition ~�0 ¼ ~� ¼ aini at zini ¼1000 for the differential equation (11). The resulting � ¼ ~�as a function of lna is shown in Fig. 1. At the second step,substituting Eqs. (16) and (17) into Eq. (13), we obtain�m

as a function of lna from the resulting differential equation.Note that cold dark matter can decay to quintessencethrough interaction, �m is unnecessary to be 1 at high

HOW TO DISTINGUISH DARK ENERGYAND MODIFIED . . . PHYSICAL REVIEW D 78, 023011 (2008)

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redshift. So, for demonstration, we choose the initial con-dition �mðz ¼ ziniÞ ¼ 0:995 at zini ¼ 1000 for the differ-ential equation of �m. Different values of �mðz ¼ ziniÞonly mean different displacements of ��0 and the dimen-sionless potential U � �2V=H2

0 at zini. The resulting �m

as a function of lna is also shown in Fig. 1. Then, followingthe prescription proposed in Sec. II, it is straightforward toobtain ��0, Q, U � �2V=H2

0 , and �� as functions of lna,while for demonstration we choose the negative branch for��0, and choose �0 ¼ 0 when we get ��. They are also

FIG. 1. � ¼ ~�, �m, ��0, Q, U � �2V=H20 , and �� as functions of lna. See text for details.


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shown in Fig. 1. Once we obtainQ,U, and �� as functionsof lna, it is easy to find Q and U as functions of ��. Theresults are shown in Fig. 2.

In short, here we have faithfully followed the recon-struction method proposed in Sec. II and successfullyconstructed an interacting quintessence model whichshares both the same expansion history and growth history

with the DGP model whose single model parameter ~�rc ¼0:170which is the best fit to the 192 SNIa data compiled byDavis et al. [10] for example. The potential Vð�Þ andcoupling Qð�Þ which are required to describe the recon-structed interacting quintessence model have been pre-sented in Fig. 2 through the equivalent Uð��Þ andQð��Þ. There are no analytical expressions for them andinstead the reconstructed Uð��Þ, Qð��Þ are given innumerical forms.

IV. CONCLUSION AND DISCUSSION

In summary, we proposed a general approach to build aninteracting quintessence model which shares both the sameexpansion history and growth history with the modifiedgravity model. Therefore, the cosmological observationsmight be unable to distinguish between them, unless othercomplementary probes beyond the ones of cosmic expan-sion history and growth history are used. As an example,we also explicitly demonstrated the interacting quintes-sence model which is indistinguishable with the DGPmodel in this sense. In fact, this proposed prescription isalso valid for other modified gravity models, such as fðRÞtheories, braneworld-type models, scalar-tensor theories(including Brans-Dicke theory), TeVeS=MOND models,and so on [47–49]. Of course, one can also extend theinteracting quintessence model to other interacting darkenergy models.

In this paper, without invoking nontrivial dark energyclustering (e.g. nonvanishing anisotropic stress), we findthat the interaction between dark energy and dark mattercould also make dark energy model and modified gravity

model indistinguishable. How can we break this degener-acy? Firstly, We should carefully check the evidences ofthe interaction between dark energy and dark matter [32–34]. If this interaction does not exist, the combined probesof cosmic expansion history and growth history might beenough to distinguish between dark energy and modifiedgravity. It is worth noting that the current observationalconstraints on the interaction between dark energy anddark matter in the literature (e.g. [32–34,50]) consideredother interaction forms (e.g. / H�m,H�de orH�tot) whichare different from the one of the present work [cf. Eqs. (3)and (4)] and hence cannot be used to compare with theinteraction considered here. Therefore, it is of interest toconsider the observational constraints on the type of inter-

actions / Qð�Þ�m_� in future works. Second, to break the

degeneracy, the other complementary probes beyond theones of cosmic expansion history and growth history aredesirable. For instance, these complementary probes mightinclude local tests of gravity, high energy phenomenology,and nonlinear structure formation. Third, in addition to thelinear matter density contrast �ðzÞ, one can also test themetric perturbations � and � by using the relationshipbetween gravitational lensing and matter overdensity [29].For the interacting quintessence model � ¼ �, whereasfor the DGP model � � �. Thus, it is possible to distin-guish between them by using delicate measurements. Weconsider that this issue deserves further investigations andbelieve that a promising future is awaiting us.

ACKNOWLEDGMENTS

We thank the anonymous referee for comments, whichhelped us to improve this manuscript. We are grateful toProfessor Rong-Gen Cai, Professor Pengjie Zhang,Professor Bin Wang, and Professor Xinmin Zhang forhelpful discussions and comments. We also thank MinziFeng, as well as Nan Liang, Rong-Jia Yang, Wei-Ke Xiao,Pu-Xun Wu, Jian Wang, Lin Lin, Bin Fan, and Yuan Liu,for kind help and discussions. We acknowledge partial

FIG. 2. Q and U � �2V=H20 as functions of ��.


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funding support from China Postdoctoral ScienceFoundation, and by the Ministry of Education of China,Directional Research Project of the Chinese Academy of

Sciences under Project No. KJCX2-YW-T03, and by theNational Natural Science Foundation of China underProject No. 10521001.

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