Towards a fully consistent parameterization of modified gravity

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by  Tessa Baker

Towards a fully consistent parameterization of modified gravity
Tessa Baker∗
Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
Pedro G. Ferreira†
Astrophysics and Oxford Martin School, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
Constantinos Skordis‡
School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD,UK
2 Joe Zuntz§
1
Astrophysics and Oxford Martin School, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK
0
Dept. of Physics & Astronomy, University College London, WC1E 6BT, UK
2

n Thereis adistinctpossibility thatcurrentand futurecosmological datacan beusedtoconstrain
a Einstein’stheoryofgravityontheverylargest scales. Tobeabletodothisinamodel-independent
J way,it makessensetoworkwith ageneral parameterization ofmodified gravity. Suchanapproach

1 wouldbeanalogoustotheParameterizedPost-Newtonian(PPN)approachwhichisusedonthescale
1 oftheSolarSystem. Afewsuchparameterizationshavebeenproposedandpreliminaryconstraints
havebeenobtained. Weshowthatthemajorityofsuchparameterizationsareonlyexactlyapplicable

] inthequasistaticregime. Onlargerscalestheyfailtoencapsulatethefullbehaviouroftypicalmodels
O
currently under consideration. We suggest that it may be possible to capture the additions to the
C ‘Parameterized Post-Friedmann’ (PPF) formalism bytreating them akin to fluid perturbations.
.
h
p
I. INTRODUCTION and luminosity at high redshift with supernovae Ia [6].
-
o With such data in hand it is possible to test cosmolog-
r ical models and constrain their parameters with some
It is possible that we live in a Universe in which more
t
s than96%oftheenergyandmatterdensityisintheform precision. With the forthcoming experiments currently
a on the drawing board [7–9], great things are expected.
[ of an exotic dark substance. The conventional view is
thatroughlyaquarterofthis obscuresubstanceisinthe In particular, there is a hope that it may be possible to

2 distinguish between the two paradigms: the dark sector
form of dark matter and the remainder is in the form
v versus modified gravity.
of dark energy. Theories abound that propose explana-
1
9 tions for dark matter and dark energy and there is an The situation in cosmology is reminiscent of that in
4 active programme of research attempting to understand General Relativity in the late 1960’s and early 1970’s.
0 and measure them. Then,Einstein’stheorywasundergoingagoldenagewith
7. Itmayalsobepossiblethatourunderstandingofgrav- discoveriesin radio and X-ray astronomy,as well as pre-
0 ityislacking,andthatEinstein’stheoryofGeneralRela- cision measurements in the Solar System and beyond,
1 tivity(andmorespecifically,theEinsteinfieldequations) making it increasingly relevant. As a result, a plethora
1 are not entirely applicable on cosmological scales. The of alternative theories of gravity were proposed which
v: pastdecadehasseenunprecedentedgrowth,fromahand- could all in principle be tested (and ruled out) by ob-
i ful to a veritable menagerie of possible modifications to servations [10–12]. Out of this situation a phenomeno-
X
gravity that may be perceived as a fictitious dark sector logical model of modified gravity emerged, the Param-
r [1]. eterized Post-Newtonian (PPN) approximation [13–16],
a
The proliferation of theories of modified gravity which could be used as a bridge between theory and ob-
couldn’t have come at a better time. Observational cos- servations. In other words, from observations it is possi-
mologyhasenteredwhatsomehavecalledaneraof‘pre- ble tofind the constraintsonthe parametersinthe PPN
cision cosmology’. Hubristic as such a point of view approximation. The constraints are model-independent.
might be, it is certainly true that cosmology is being Fromanygiventheoryitisthenpossibletocalculatethe
inundated by data, from measurements of the Cosmic correspondingPPN parameters and find if they conform
MicrowaveBackground(CMB) [2,3], galaxysurveys[4], to observations. The PPN approximation is sufficiently
weak lensing surveys [5] and measurements of distance general that it can encompass almost all modified theo-
ries of gravity that were then proposed.
Clearly something like the PPN approach is desirable
in cosmology. Given the rapid increase in the number of
∗Electronicaddress: [email protected]
modified theories of gravity,it would make sense to con-
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] structaparameterizationthatcouldserveasabridgebe-
§Electronicaddress: [email protected] tween theory and observation. Observers could express
2
their constraints in terms of a set of convenient parame- Φ)aregravitationalpotentialsandγPPN isaPPNparam-
ters; theorists could then make predictions for these pa- eter,equivalenttooneoftheolderEddington-Robertson-
rameters and check if their theories are observationally Schiff parameters.
viable. Instead of performing many constraint analyses Thereareafewpropertieswhichareofnoteinthisex-
on individual theories, one could run just a single con- pression. Firstofall,theparameterizationisconstructed
straint analysis on the parameterized framework. With around a solution of the Einstein field equations, the
a dictionaryoftranslationsbetweentheoriesandthe pa- Schwarzschild solution (with γPPN =1) – the field equa-
rameterization in hand, these general constraints could tions do not come into play. Second (and this isn’t obvi-
be immediately applied to any particular theory. ousfromtheexpressionsabove),theparameterγPPN only
Anotherkeyadvantageofaparameterizedapproachis depends on parameters in the theory and not on inte-
that it allows one to explore regions of theory space for gration constants or ‘environmental’ parameters such as
which the underlying action is not known. For example, the central mass. This means that, given a theory, it
in VIBwewillseehowtheLagrangianf(R)isrelatedto is possible to predict γPPN solely in terms of fundamen-
§
our framework ‘parameters’ (which are really functions, tal parameters of the theory (i.e. the parameters in the
not single numbers – see below). Cosmological data will action). Finally, we see that the mismatch between the
excludecertainregionsofparameterspace. Ifanewform gravitationalpotentials can be expressed as [14]
of f(R) is proposed in the future, it should be a quick
operationto see whether it falls into the excluded region Φ Ψ=ζPPNΦ (2)
−
-eventhoughthatparticularLagrangianwasnotknown
at the time that the constraint analysis was performed. with ζPPN = (γPPN−1)/γPPN often called the gravitational
In this paper we will discuss the requirements of how slip. General Relativity is recovered when ζPPN =0.
Theideaofapplyinganequationoftheformofeqn.(2)
to parametrize modifications to gravity on cosmological
to cosmology emerged from the work of Bertschinger in
scales. It develops the principles first put forwardin [17]
[19]. Bertschingershowedthatonlargescalesitwaspos-
and explores how they may be applied more generally.
sible tocalculate the evolutionofΨ andΦ using onlyin-
The layout of this paper is as follows. In II we discuss
§ formationaboutthe backgroundevolutionandassuming
theideabehindthePPNapproachandshowthatitcan’t
a closure relationship between the two potentials. The
be imported wholesale into cosmology. We then briefly
simplest assumption is a closure relation of the form of
look at attempts at parametrizing gravity and point out
eqn.(2), but in no way was it implied that this would be
their limitations. In III we discuss a possible formal-
§ arealistic relationshipthatwouldbe validinthe general
ism in detail – we co-opt the name Parameterized Post-
space of theories of modified gravity.
Friedmann approach from [18] – and argue that it may
Nevertheless, over the last few years the simplified
be sufficiently generalto encompassa broadclassofthe-
equationforgravitationalsliphasbeenadoptedasagen-
ories. In IV we construct the hierarchy of equations
§ eralparameterizationwhichshouldbevalidincosmology
that shouldbe satisfied inthe case in which there are no
[20–22]. It has been shown to be valid in a few cases, in
extra fields contributing to the modifications to gravity.
the quasistatic regime (i.e. on small scales), and explicit
In V we discuss the more general case with extra fields
§ expressionshavebeenfoundforζintermsoffundamental
and how this affects the relations between the different
parametersofthosetheories(someexamplesarecollected
coefficients. In VI we focus on four modified theories of
§ in[1]). Suchaparameterizationhasbeenextendedtoin-
gravity and analyse how they fit into the formalism that
clude another parameter, a modified Newton’s constant
we are proposing. Finally, in VII we summarize the
§ G , which may differ from G . The method is then to
state of play of the parameterization we are proposing. eff 0
use eqn.(2) and a modified Poissonequation,
II. THE CONVENTIONAL APPROACH k2Φ=4πGeffa2 ρi∆i (3)
−
i
X
Theplanistoconstructaparameterizationthatmight
(whereρ istheenergydensityoffluidiand∆ istheco-
mimic the PPN approach on cosmological scales. It i i
movingenergydensity)tomodifytheevolutionequations
is therefore useful to look very briefly at the PPN ap-
for cosmological perturbations. A modified Einstein-
proach, which proceeds as a perturbative expansion in
Boltzman solver is then used to calculate cosmological
v/c (though we set c=1 in what follows). Consider the
observables. The two parameters (G , ζ) have been
modified, linearized Schwarzschildsolution: eff
adopted more generally and have been used to find pre-
ds2 = (1 2Ψ)dt2+(1 2Φ)dr2+r2dΩ2 liminary constraints on modified gravity theories by a
− − −
number of groups [21–24].
G M
0
Ψ = Clearly such an approach to parametrizing modified
r
gravity has some significant differences with the PPN
Φ = γPPNΨ (1)
approach. For a start, modifications are applied to the
whereG isNewton’sconstant,M isthecentralmass,Ω field equations and not to specific solutions of Einstein’s
0
is the two-dimensional volume element of a sphere, (Ψ, field equations. This is understandable – the solutions
3
of interest in cosmology are not only inhomogeneous Bianchi identity Gµ =0 implies the relation:
∇µ ν
but time-varying, unlike the incredible simplicity of the
Schwarzschild solution that arises due to Birkhoff’s the- E˙ + (E +3E )=0 (7)
F F R
H
orem. Also, unlike in PPN, the parameters at play – ζ
and Geff – will not only depend on fundamental param- Assumingthattheconservationlaw∇µTνµ =0holdssep-
eters of the theory but also on the time evolution of the aratelyforordinarymatterandtheeffectivedarkfluid,X
cosmologicalbackground. andY mustberelatedbytheequationX˙ +3 (X+Y)=
H
0.
Ideally, any time-dependence in the parameterization
Continuinginthisvein,ourgoalistowritethelinearly
willbesimplyrelatedtobackgroundcosmologicalquanti-
perturbed Einstein equations as:
ties(likethescalefactor,energydensitiesoranyauxiliary
fields that are part of the modifications) and not depen-
δG = 8πG a2δT +a2δU (8)
dent upon the time evolution of Φ, Ψ or any other per- µν 0 µν µν
turbation variables. For such a requirement to be possi-
In general the tensor δU will contain both metric per-
µν
bleitisessentialthatanyparameterizationissufficiently
turbationsandextra degreesoffreedom(hereafterd.o.f)
general to encompass a broad range of theories. As we
introduced by a theory of modified gravity. We can sep-
will showin this paper, parameterizationsusing eqns.(2)
arate δU into three parts: i) a part containing only
µν
and(3)aresimplynotgeneralenoughtocapturethefull
metric perturbations,ii) a partcontaining perturbations
rangeofbehaviourof modifiedtheories ofgravity. It has
to the extra d.o.f., iii) a partmixing the extra d.o.f. and
beenarguedthatsuchaparameterizationcanbeusedas
perturbations to the ordinary matter components:
a diagnostic; that is, for example, a non-zero measure-
ment of ζ might indicate modifications of gravity [25].
δU =δUmetric(Φˆ,Γˆ...)+δUdof(χ,χ˙,χ¨...)+δUmix(δρ...)
This may be true, but such a measurement cannot then µν µν µν µν
be used to go further and constrain specific theories. It (9)
would be more useful to build a fully consistent parame-
The argument variables in this expression will be intro-
terization which can be used as a diagnostic and can be
duced shortly. We have written the Einstein field equa-
linkedtotheoreticalproposals. Thepurposeofthispaper
tions such that Tµ contains only standard, uncoupled
istotakethefirststepstowardssuchaparameterization. ν
mattertermsandhenceobeystheusual(perturbed)con-
servation equations, δ( Tµ)=0. As a result Uµ must
∇µ ν ν
obeyitsownindependentconservationequations,sothat
III. THE FORMALISM atlinearorderwehaveδ( Uµ)=0. Wewillusethefol-
lowing notation to denote∇cµomνponents of δUµ from here
ν
When considering modified gravity theories it can be onwards:
helpful to cleanly separate the non-standard parts from
thefamiliartermsthatariseinGeneralRelativity(hence- U∆ = −a2δU00, ∇~iUΘ =−a2δUi0 (10)
forth GR). We can always write the modifications as a 1
U = a2δUi, D U =a2(Ui δUkδi)
anadditionaltensorappearingintheEinsteinfieldequa- P i ij Σ j − 3 k j
tions, i.e.
where D = ~ ~ 1/3q ~2 projects out the longitu-
ij i j ij
Gµν = 8πG0a2Tµν +a2Uµν (4) dinal, traceles∇s p∇art−of δUνµ∇and qij is a maximally sym-
metric metric of constant curvature K. The definition
The diagonal components of the tensor Uµν are equiva- of UΣ. In the case of unmodified background equations
lent to aneffective dark fluid with energy density X and perturbed conservation equation for Uνµ gives us the fol-
isotropic pressure Y (where the constants have been ab- lowing two constraint equations at the linearized level
sorbed). The zeroth-order Einstein equations are then: [17]:
1
EF ≡ 3H2+3K =8πG0a2 ρi+a2X (5) U˙∆+HU∆−∇~2UΘ+ 2a2(X +Y)(β˙ +2∇~2ǫ)
Xi + UP = 0(11)
ER ≡ −(2H˙ +H2+K)=8πG0a2 Pi+a2Y (6) U˙ +2 U 1U 2~2U +a2(X +HY)Ξ = 0(12)
Xi Θ H Θ− 3 P − 3∇ Σ
where =H/a is the conformal Hubble parameter and where the metric fluctuations β, ǫ and Ξ are defined in
H
K is the curvature. We will use E and E as defined eqn.(13).
F R
above throughout this paper. For future use we define Inthis paperwewillinitiallypresentgeneralformsfor
E =E +E . Thesummationsintheaboveexpressions the construction of metric-only δU that satisfy equa-
F R µν
are taken over all conventional fluids and dark matter, tions (11) and (12), then impose the restriction that
and dots denote derivatives with respect to conformal the field equations can contain at most second-order
time. In this paper we will largely adhere to the defini- derivatives. Gravitationaltheoriescontainingderivatives
tionsandconventionsusedin[17]. Wealsonotethatthe greater than second-order are generally disfavoured as
4
they typically result in instabilities or the presence of In terms of these variables the perturbed Einstein equa-
ghost solutions [1, 26, 27]. However, we note that some tions are [17]:
special cases of higher-order theories are acceptable e.g.
f(R) theories [28] (see VIB). Hence we start with the E = 8πGa2 ρ δ +U (19)
§ ∆ i i ∆
general case in order to indicate how our results may be i
X
extended to higher-derivative theories [29]. E = 8πGa2 (ρ +P )θ +U (20)
Θ i i i Θ
Therequirementofsecond-orderfieldequationsmeans
i
that U and U can only contain first-order derivatives X
∆ Θ
with respect to conformal time, as can be seen from EP = 24πGa2 ρiΠi+UP (21)
eqns.(11) and (12). The specific implications this has i
X
depends on which of the tensors in eqn.(9) are present. E = 8πGa2 (ρ +P )Σ +U (22)
Σ i i i Σ
In IVwewillexplorethestructureoftheorieswithonly
§ Xi
metric perturbations, whilst theories with extra degrees
of freedom will be presented in V. In Appendix B we For simplicity we will hereafter consider only the case of
display formulae for generating c§onstraint equations in a universe with zero spatial curvature, K =0.
an arbitrary-order theory of gravity with no additional
degrees of freedom.
Wewritetheperturbedlineelement(forscalarpertur- IV. THE GENERAL PARAMETERIZATION -
bations only) as: NO EXTRA FIELDS
ds2 = a2(1 2Ξ)dt2 2a2(~ ǫ)dtdx
− − − ∇i A. General case - unmodified background
1
+ a2 1+ β q +D ν dxidxj (13)
ij ij
3
(cid:20)(cid:18) (cid:19) (cid:21) Let us begin with the simplest case by applying two
It will prove useful to define the gauge-variant combina- restrictions: i) We consider the case of modifications to
tion: gravity that appear only at the perturbative level, that
is, they maintain the background equations of GR for
V =ν˙ +2ǫ (14)
a Friedmann-Robertson-Walker metric; ii) there are no
We also define the gauge-invariantpotentials: new d.o.f. present in the theory, so δU contains only
µν
1 1 metricperturbations. Wewillrelaxrestrictioni)in IVD
Φˆ = (β ~2ν)+ V (15) §
−6 −∇ 2H and restriction ii) in V. We will see shortly that the
§
1 1 treatment presented in this subsection is also applicable
Ψˆ = Ξ V˙ V (16)
to ΛCDM, because the X + Y terms in eqns.(11) and
− − 2 − 2H
(12) vanish for a cosmological constant. The agreement
Φˆ and Ψˆ are equivalent to the Bardeen potentials Ψ
and Φ respectively. Note that Ψˆ contains a sec−ondH- between an exact ΛCDM background and current data
A means that theories obeying restrictions i) and ii) are
order time-derivative. In the first four sections of this
of particular interest, even though they correspond to a
paperwewillfrequentlyusealinearcombinationofthese
limited region of theory space.
variables that remains first-order in time derivatives of
The requirement of gauge form-invariance places
perturbations,duetoacancellationbetweentheV˙ terms:
strong restrictions on the forms that δU can take [17].
µν
Γˆ = 1 Φˆ˙ + Ψˆ (17) We will postpone a detailed discussion of these restric-
k H tionsuntil IVD,wheretheywillbeausefultoolinguid-
From (cid:16)V onward(cid:17)s we specialise to the conformal Newto- ing us to a§llowed combinations of metric perturbations.
nian g§auge, and hence revert to the familiar potentials In this subsection it suffices to point out that the stan-
Φ and Ψ. We introduce a shorthand notation for the dardEinsteinfieldequationsofGRareofcoursealready
components of δG in exactanalogyto that introduced gauge form-invariant; so any additive modification like
µν
fsoidreδsUoµfνt,hie.ep.eErt∆ur=be−daE2iδnGst00eientce.quHaetrieoanfstewritllhbeeledfet-nhoatnedd δdUerµνtompursetsebreveindtheepeinndvaenritalyncgeaoufgethfoermwh-ionlveareixapnrteisnsioonr-.
by: This property is a directconsequence of the fact that we
have not yet modified the backgroundequations. In this
3
E∆ = 2(~2+3K)Φˆ 6 kΓˆ EV case the only objects that can be present in the tensor
∇ − H − 2H
δU are the gauge-invariantmetric potentials Φˆ and Γˆ.
1 µν
EΘ = 2kΓˆ+ 2EV So, we can construct the tensor δUµν from series of
all the possible derivatives of Φˆ and Γˆ. This structure
dΓˆ
E = 6k +12 kΓˆ 2(~2+3K)(Φˆ Ψˆ) should be general enough to encompass any metric the-
P
dτ H − ∇ − ories,where the actionis constructed purely fromcurva-
3
3EΨˆ + E˙ 2 E V ture invariants, e.g. f(R) gravity, Gauss-Bonnet gravity
R R
− 2 − H [30]andLovelockgravity[31]. Ifwewishtoparameterize
E = Φˆ Ψˆ (cid:16) (cid:17) (18) only second-ordertheories then we will need to truncate
Σ
−
5
these series at N = 2, as discussed in section III. The this a¨ term by adding a term proportional to EV, but
components of U are given by: this would break the gauge-invariance of the perturbed
µν
Einsteinequations. Wewillseelaterthatmodificationof
N−2
the background equations allows us to add an EV term
U = k2−n A Φˆ(n)+F Γˆ(n) (23)
∆ n n withoutviolatinggauge-invariance,whichinturnsmeans
nX=0 (cid:16) (cid:17) that Γˆ can be present in U∆ and UΘ.
N−2 Usingeqns.(11)and(12)wefindthatsettingF =I =
U = k1−n B Φˆ(n)+I Γˆ(n) (24) 0 0
Θ n n 0 forces J = K = 0 also. Then, for the second-order
1 1
nX=0 (cid:16) (cid:17) case, the remaining terms in δUµν are:
N−1
UP = k2−n CnΦˆ(n)+JnΓˆ(n) (25) U∆ = A0k2Φˆ
nX=0 (cid:16) (cid:17) UΘ = B0kΦˆ
N−1
UΣ = k−n DnΦˆ(n)+KnΓˆ(n) (26) UP = C0k2Φˆ +C1kΦˆ˙ +J0k2Γˆ
nX=0 (cid:16) (cid:17) U = D Φˆ + D1Φˆ˙ +K Γˆ (27)
Σ 0 0
k
The coefficients A -K are functions of the scale factor
n n
a,wavenumberk andbackgroundquantities suchasρ˙i – The constraint equations are given in Table I, indicat-
forthesakeofclaritywewillsuppressthesedependencies ingthe terms andBianchiidentity fromwhichtheyarise
throughout. The factors of k ensure that the coefficient (B1 eqn.(11), B2 eqn.(12)). These expressionscan
⇒ ⇒
functions are dimensionless. be generated using the formulae in Appendix B. We can
Let us take a moment to explain the upper limits on see immediately that the Γˆ terms in U and U vanish,
P Σ
thesummationsineqns.(23)-(26). Φˆ andΓˆ arefirst-order leavingδU expressedentirelyinterms ofΦˆ andΦˆ˙. We
µν
intime derivatives(seeeqns.(15)and(17)). U is differ-
∆ have two free functions remaining, which we will choose
entiated in eqn.(11), so truncating the series in eqn.(23) to be D and D . Eliminating C from the two Φˆ˙ con-
at Φˆ(N−2) gives field equations containing time deriva- 0 1 1
straints gives (where = /k):
tivesoforderN.UΘ istreatedanalogouslytoU∆. AsUP Hk H
and UΣ are not differentiated in the components of the 1
Bianchi identity, the series in eqns.(25) and (26) are al- HD1 = (A0+3 kB0) (28)
k −2 H
lowedto extend one orderhigherthan those ineqns.(23)
and (24). Thecombinationontheright-handsideappearswhenwe
We substitute our forms for U , U , U and U into formthe(Fourier-space)Poissonequationfromeqns.(19)
∆ Θ P Σ
the components of the Bianchi identity (11) and (12). Φˆ and (20), where it acts to modify the value of Newton’s
and Γˆ are non-dynamical fields and so will not evolve gravitationalconstant:
in the absence of source terms. Yet when we perform
G
the substitution, the Bianchi identity appears to give us k2Φˆ =4π 0 a2 ρ ∆ (29)
i i
two evolution equations for Φˆ and Γˆ. The only way this − 1 g˜
− i
can be avoided is if the coefficients of each term Φˆ(n) X
andΓˆ(n) vanishindividually,whichprovidesuswithcon- where∆i =δi+3 (1+wi)θi isagauge-invariantmatter
H
straintequationsonthefunctionsA -K (thisprocedure perturbation and
n n
will be clarified with an example shortly). Each compo-
1
nentofthe Bianchiidentity resultsinNconstraintequa- g˜= (A +3 B ) (30)
0 k 0
tions from each of the Φˆ(n) terms and Γˆ(n) terms, and −2 H
eqns.(23)-(26)contain8N-4coefficientfunctionsintotal. The sum in eqn.(29) is over all known fluids and dark
Hence we have 4N 4 free functions with which to de-
matter, and G denotes the canonical value of Newton’s
− 0
scribe the theory. constant. From here on we will replace D /k in eqn.(27)
1
byg˜/ toremindusoftheconnectionbetweenthemod-
H
ifications to the slip relation and the Poisson equation.
B. Second-order case - unmodified background
We will also replace D by ζ to distinguish it from the
0
other coefficient functions, which can all be expressed in
In Appendix B we give formulae for generating the termsofg˜andζ usingtheconstraintequations. Wecon-
constraint equations of an arbitrary-order theory with tinue to suppress the arguments of g˜ and ζ.
unmodified background equations. We will now explic- The effective gravitational constant appearing in the
itly present the second-order case, which corresponds to Poisson equation is G =G /(1 g˜). The traceless
eff 0
setting N=2 in eqns.(23)-(26). In a generalcase this will space-space component of the Ein−stein equations be-
give us four free functions. However, if the background comes:
equationsareunalteredthenwemustsetF =I =0be-
0 0
cause Γˆ contains a second-order conformal time deriva- Φˆ Ψˆ =8πG (ρ +P )Σ +ζΦˆ + g˜Φˆ˙ (31)
0 i i i
tive of the scale factor. One might consider cancelling −
i H
X
6
Origin Constraint equation Inparameterizationsequivalentto(Q,ηslip)thedegener-
1 [B1] Φˆ A˙0+HA0+kB0+HC0 =0 acy is Q(1+1/ηslip), so for lensing applicationsit makes
2 [B1] Φˆ˙ A0+HkC1 =0 stoentsheetdoegdeenfienreacnyewdirpeacrtaiomne[t3e2r–s3a4l]o.nIgnatnhde(pg˜e,rζp)enpdaricaumlaer-
3 [B1] Γˆ J0 =0 terizationadegeneracyremains. Thedominantcontribu-
4 [B2] Φˆ B˙0+2HB0− 13kC0+ 23kD0 =0 tions to lensing signals come from quasistatic scales, on
5 [B2] Φˆ˙ B0− 31C1+ 32D1 =0 whichtime derivatives ofperturbations canbe neglected
6 [B2] Γˆ 2K0−J0 =0 (seelaterforafullerdiscussion). Thedegeneracyisthen:
TABLE I: Table of the constraint equations for the second-
(2 ζ)
ordermetrictheoryspecifiedin§IVB.Thesecanbegenerated k2(Φ+Ψ)=4πG a2 ρ δ − (34)
0 i i
− 1 g˜
using theformulae in AppendixB.
i −
X
It seems that neither of the two parameterizations pre-
The anisotropic stress perturbation Σi is automatically sented so far are optimal for weak lensing constraints.
gauge-invariant,but negligiblefor standardfluids atlate
times. The above expression echoes its PPN equivalent,
eqn.(2); but note that, as discussed in II, ζ is a func-
§
tion of background quantities (which potentially intro-
C. Why neglecting g˜ in the slip relation implies a
ducetime-andscale-dependence),whereasζPPNdepended higher-derivative theory
onlyuponfundamentalparametersofagravitationalthe-
ory.
We have seen in the previous section that in a metric-
Other authors have made numerous different choices
based second-order theory of modified gravity the most
for the two free functions of a second-order theory; a
generalformofthegravitationalslipshouldbeexpressed
useful summary of some of these is provided by [22] . A
in terms of the gauge-invariant potential Φˆ and its first
common choice is to introduce a function Q = G /G ,
eff 0
related to our g˜ by Q = (1 g˜)−1 [21] (though dif- derivativewithrespecttoconformaltime. Twofreefunc-
− tions ζ andg˜were usedas the coefficients of these terms
ferent notation is in no short supply) . The relation-
respectively, where g˜ resulted in a modification to New-
ship between the two potentials is often parameterized
as Φˆ =η (a,k)Ψˆ in the spirit of the PPN parameter ton’sgravitationalconstantinthe Poissonequation. Us-
slip ing a single function to relate Φˆ and Ψˆ is equivalent to
γ [13]. It might be felt that by introducing yet an-
PPN
setting g˜ = 0 (see eqn.(31)), which is inconsistent with
otherparameterizationofPPFweareaddingtothis dis-
with allowing a second free function to modify Newton’s
array. However,inthenextsubsectionwewillarguethat
constant. Making the choice g˜=0 uses up one degree of
a two-functionslip relationsuch as eqn.(31) is needed to
freedom,leavingusonlyasinglefreefunctionwithwhich
avoidimplicitly introducinghigher-orderderivativesinto
to describe the system.
a purely metric theory.
Writing the relationship between the two gauge- The above reasoning is set within the confines of a
invariant potentials as Φˆ =η (a,k)Ψˆ implies that the second-order theory. We will now show that using a
slip
spatialoff-diagonalcomponentoftheEinsteinfieldequa- single function to relate Φˆ and Ψˆ whilst maintaining
tions is: Geff =G0 is equivalent to invoking a higher-derivative
6
theory of gravity. To do this, let us consider the form
Φˆ Ψˆ = 1 1 Φˆ (32) that the tensor δUµν would take in a third-order theory.
− − η Its components would be:
(cid:18) slip(cid:19)
Comparing the above equation with eqn.(31) implies: U = A k2Φˆ +A kΦˆ˙ +F k2Γˆ
∆ 0 1 0
η−1 =1 ζ g˜ dlnΦˆ (33) UΘ = B0kΦˆ +B1Φˆ˙ +I0kΓˆ
slip − − H dτ UP = C0k2Φˆ +C1kΦˆ˙ +C2Φ¨ˆ +J0k2Γˆ+J1kΓˆ˙
Now ηslip has an environmental dependence, which is U = D Φˆ + D1Φˆ˙ + D2Φ¨ˆ +K Γˆ+ K1Γˆ˙ (35)
problematic. Wewouldrequiredetailedknowledgeofthe ∆ 0 k k2 0 k
environment in which we wish to test a theory a priori,
andthe PPFfunctions wouldneedtobe recalculatedfor The constraint equations for this system are given in
numerous different situations. Unless Φˆ˙ =0, the param- Table II. Wewillcontinuetodefinethecombinationthat
eterizations in eqns.(31) and (32) do not have a simple modifiesG asg˜= 0.5(A +3 B ),butnotethatthis
0 0 k 0
− H
equivalence. is no longer equal to D as it was in the second-order
k 1
H
A degeneracy arises between g˜ and ζ when comparing case. Consider the case where we set D = D = K =
1 2 0
todatafromweakgravitationallensing,whichprobesthe K = 0, that is, we use a single function to relate Φˆ
1
combination Φ+Ψ in the conformal Newtonian gauge. and Ψˆ. Through linear combinations of the constraints
7
Origin Constraint equation in eqns.(35) there are no nonlocal terms present in the
1 [B1] Φˆ A˙0+HA0+kB0+HC0 =0 gravitational field equations, so we should not be sur-
2 [B1] Φˆ˙ A˙1+HA1+kA0+kB1+HC1 =0 prised that the Lovelock-Grigore theorem prevents us
from obtaining a second-order theory. In VI we will
3 [B1] Φ¨ˆ kA1+HC2 =0 meettheorieswhichevadethe Lovelock-Grig§oretheorem
4 [B1] Γˆ F˙0+HF0+kI0+HJ0 =0 in a number of different ways: by introducing new d.o.f.
5 [B1] Γˆ˙ kF0+HJ1 =0 (e.g. scalar-tensor theory), higher-order field equations
6 [B2] Φˆ B˙0+2HB0− 13kC0+ 23kD0 =0 f(R) gravity , or throughnonlocality and extra dimen-
7 [B2] Φˆ˙ B˙1+kB0+2HB1− 31kC1+ 23kD1 =0 s(cid:0)ions (DGP).(cid:1)
8 [B2] Φ¨ˆ B1− 31C2+ 32D2 =0
9 [B2] Γˆ I˙0+2HI0− 13kJ0+ 32kK0 =0 D. Cases with ‘XY’ backgrounds
10 [B2] Γˆ˙ I0− 13J1+ 23K1 =0
The previous examples have all assumed that the
TABLE II: Table of the constraint equations for the third-
background field equations are those of a Friedmann-
ordermetrictheoryspecifiedin§IVC.Thesecanbegenerated
Robertson-Walkermetricplusstandardcosmologicalflu-
using theformulae in AppendixB.
ids. We now relax this assumption and consider theories
which modify the Einstein field equations at both the
in Table II we derive the expressions: backgroundandperturbativelevels. Itiswell-knownthat
anymodificationtobackground-levelfieldequationsisin-
A1+3 kB1 =0 (36) distinguishablefromtheeffectsofadarkfluid[37];hence
H
F +3 I =0 (37) we can write any background equations as the standard
0 k 0
H
FRW ones with an additional energy density and pres-
3 1 ˙ 1 3E
g˜= B 2 + Hk = B +k2 (38) sure, see eqns.(5) and (6). We will refer to such theories
2 1 Hk 3 − k ! 2k2 1(cid:18) 2 (cid:19) as having ‘XY backgrounds’.
AnyextensiontoGRmustpreservethepropertyofdif-
The first two of these expressions are the combinations
feomorphisminvariance. Invarianceunderpassivediffeo-
that appear when we form the Poisson equation. They
morphisms corresponds to the familiar principle of gen-
indicate that the potential additive modifications pro-
eral covariance. Applying a passive diffeomorphism will
portionaltoΦˆ˙ andΓˆ disappear;theformatofeqn.(29)is generally result in field equations which look different
retained. Eqn.(38) shows that we can only have a modi- to those in the old co-ordinate system. In contrast, in-
fication to the effective gravitational constant if B1 =0, variance under active diffeomorphisms requires that the
6
and so from eqn.(36) A1 =0 also. Using the third equa- actual form of field equations remains unchanged by a
6
tioninTableII, C2 =0inthiscase. Henceweareforced gauge transformation. In Table III we list the gauge
to include Φˆ˙ terms i6n U and U , and a Φ¨ˆ term in U . transformationsforrelevantvariables. Inpracticalterms,
∆ Θ P
SinceΦˆ containsafirst-ordertimederivativealready(see gauge form-invariance means that the extra terms that
eqn.(15), the U˙ ineqn.(11) willresultinfieldequations appear under a gauge transformation must cancel each
∆
other(usingidentitiesfromthezero-orderfieldequations
containing third-order time derivatives - a higher-order
ifneedbe). Thisplacestightrestrictionsonourformfor
gravitationaltheory.
δU .
This result is a direct consequence of choosing D =0 µν
1
in U . Removing this constraint changes eqn.(38) to: To see how this happens in ordinary GR, consider the
Σ
linearly perturbed ‘00’ component of the Einstein equa-
g˜= 1 B 3E +k2 + D (39) tions (that is, eqn.(19) with U∆ set to zero). When
2k2 1 2 Hk 1 we apply a gauge transformation the left-hand side ac-
(cid:18) (cid:19) quires a term 3 Eξ/a. This is cancelled by the trans-
− H
which permits B =0,g˜=0, as we had in IVB. formation of δ on the right-hand side, provided that
1
The above findings ma6 ke sense within th§e context of E =EF +ER =8πGa2 iρi(1+wi),i.e. providedthat
the Lovelock-Grigoretheorem [35, 36], which states that the zeroth-order equations are satisfied. This is why
P
under the assumptions of four-dimensional Riemannian we were only able to use gauge-invariant potentials in
geometry and no additional fields, the Einstein-Hilbert δUµν in IVA and IVB: if we don’t alter the zeroth-
§ §
action(plus acosmologicalconstant)is the onlypossible orderequations,addinganythingelsebreaksgaugeform-
action that leads to local second-order field equations. invariance.
In eqn.(27) the presence of D /k in U means that this Now that we wish to consider XY backgrounds
1 Σ
parameterization implies a non-local theory. This is not this procedure no longer works, because E =
in itself problematic – nonlocal theories can arise when 8πG a2 ρ (1+w ). We must add a new term to U6
0 i i i ∆
a degree of freedom has been integrated out, or elimi- thatwillproduceapartlike 3 a2(X+Y)ξunderagauge
P aH
nated from the action using an integral solution of the transformation. Only then will the gauge-variant parts
correspondingequationofmotion. IfD =D =K =0 cancel by virtue of the zeroth-order equation.
1 2 1
8
Metric variables Fluid variables explicitly first-order in derivatives, resulting in second-
Ξ→Ξ− ξ˙ δ →δ− 3(1+w)Hξ order equations.
a a
ǫ→ǫ+ 1[ξ+Hψ−ψ˙] θ→θ+ 1ξ UΘ is treated analogously to U∆, and has the form:
a a
β →β+ 1[6Hξ−2k2ψ] Π →Π+ 1[w˙ −3wH(1+w)]ξ
ν →aν+ 2ψ aΣ→Σ UΘ =2g˜b kΓˆ+ 1EV +B0kΦˆ (43)
a 4
V →V + 2ξ (cid:20) (cid:21)
a
Combining eqns.(42) and (43), we find that the Poisson
Components of δGµν equationhasthe sameformasitdidinthecasewithun-
E∆→E∆− a3H(EF +ER)ξ EΘ→EΘ+ a1(EF +ER)ξ mcoomdbifiineadtiboanckg˜g=roun1d/2eq(Auat+ion3s, eBqn.)(2a9s)w.eWdeiddienfinIeVthBe.
EP →EP + 3aξ(E˙R−2HER) EΣ→EΣ Wewillassume th−atthem0odifiHcaktio0nstoNewton’s§grav-
itationalconstantappearinginthezeroth-orderandper-
TABLE III: Behaviour of metric and fluid variables under
turbed equations are the same, i.e. g˜ =g˜, noting that
infinitesimal diffeomorphisms generated by the vector field b
ξµ =a(−ξ,∇~iψ). Note that the shear Σ is gauge-invariant. we have not formally proved this to be the case.
OncewehavededucedtheformofU andU ,U and
∆ Θ P
U can be found using the Bianchi identities eqns.(11)
Σ
As a toy example, consider a simple theory which and (12) . Eliminating the free function A in favour of
0 (cid:0)
modifies the zeroth-order equations solely by introduc- g˜ and B , these are:
0
(cid:1)
ing time-dependence to Newton’s gravitationalconstant.
2
Followingthe notationofprevioussections,wecanwrite U = 3Φˆ k B˙ +kB L+ k2 g˜˙ + g˜
P 0 0
the sum of the modified Friedmann and Raychaudhuri H H 3 H
(cid:20) (cid:21)
equations as: +3Φˆ˙ k B +g˜ E + k2(cid:0) (cid:1)
0
H 2 3
G (cid:20) (cid:18) (cid:19)(cid:21)
E =8π1 g˜0b(a)a2 ρi(1+wi) (40) +6 g˜˙ +g˜L kΓˆ+ 1EV
− Xi H 4
(cid:20) (cid:21)
RewritingthisintheformofordinaryGR(andhereafter +6g˜(cid:0) d kΓˆ(cid:1)+ 1EV
suppressing the argument of g˜ ): Hdτ 4
b (cid:20) (cid:21)
3 k2 3
E =8πG a2 ρ (1+w )+g˜ E (41) g˜EV ˙ Eg˜ V˙ (44)
0 i i b 2 3 −H − 2 H
(cid:20) (cid:21)
i
X
fromwhichwecanidentifya2(X+Y)=g˜ E(seeeqns.(5) Φˆ E k2
b U = 2k g˜˙ + g˜ 3B +
and (6)). A possible form for U∆ is then: Σ 2 k H − 0 2 3
H (cid:20) (cid:18) (cid:19)(cid:21)
1 g˜ kΓˆ (cid:0) Φˆ˙ (cid:1)
U∆ = 6g˜b kΓˆ+ EV +A0k2Φˆ (42) − −
− H(cid:20) 4 (cid:21) H(cid:16) (cid:17) k2
where L= 2+ ˙ . (45)
Unlike the second-order example of IVB, U now con- H H− 3
∆
tains a Γˆ term. The offending seco§nd-order derivative As expected, we find that U contains only gauge-
of the scale factor present in Γˆ is cancelled by the term invariant perturbation variablesΣ. This must be the case
proportional to EV. We did not have the freedom to since all other terms in eqn.(22) are gauge-invariant, so
add such a term in the case of unmodified background there is nothing to cancel against.
equations, because V is gauge-variant. If we express the last term in U in terms of Ψˆ we
Σ
Using eqn.(42) in eqn.(19) and the transformations find that we can write the relationship between the two
given in Table III, it can be verified that the gauge- potentials as Φˆ = η (a,k)Ψˆ for this toy example. We
slip
variant parts cancel by satisfying eqn.(40). Note that have already chosen our two PPF functions to be g˜ and
the need to have gauge-invariantsecond-order equations B , so η is simply a particular combination of these:
0 slip
hastotallyfixedthefirsttermineqn.(42);allfreedomre-
sides in the gauge-invariant part of U∆ via the function (1 g˜)
η = H − (46)
A0. slip (1 g˜) g˜˙ + 3B0 E + k2
As a quick sanity check, one can verify that the field H − − 2k 2 3
equations remain second order in the conformal Newto- (cid:0) (cid:1)
The key result of this toy example is that (for purely
nian gauge. In this gauge V = V˙ = 0, Ξ = Ψ and
metric theories) δU can have a more complex form
β = 6Φ, so Φˆ and Ψˆ reduce to their familia−r coun- µν
when the background equations are not standard FRW
−
terparts Φ and Ψ that appear in the linearly perturbed
plus standardcosmologicalcomponents (baryons, CDM,
FRW metric (recall that Γˆ =1/k(Φˆ˙ + Ψ)). U is then etc.) Thehierarchyofconstraintequationsbecomesmore
∆
H
9
complexduetothenon-zeroX andY termsineqns.(11) withthesamedimensionalityasχ~. α~ ,A ,F andM are
i i i i
and (12). We can use the principles of energy conserva- functionsofbackgroundquantitiessuchasρanda;these
tion,gauge-invarianceandsecond-orderfieldequationsas dependencieshavebeensuppressedforclarity. Notethat
ashortcuttothecorrectforms;thesameresultswouldbe eqn.(47)has the formindicatedschematicallyineqn.(9).
obtainedbysolvingthehierarchyofconstraintequations ThetermM δρ˙ representsamodificationwhichdepends
1
directly. on the rate of change of the density fluctuations of or-
dinary matter. Whilst formally this term is permitted
to be present in U , we are unaware of any theory of
∆
V. THE GENERAL PARAMETERIZATION- modified gravity that results in a perturbed 00-equation
EXTRA FIELDS with a term like this. Theories employing a chameleon
mechanism introduce modifications to GR that depend
The formalism we have developed so far is only ap- uponthe environmentalmatterdensity,butnotuponits
plicable to purely metric theories. Yet, as discussed in rateofchange. WewillthereforechooseM =0inwhat
1
II, the majority of modified gravity theories introduce follows. Theδρ termineqn.(47)canbe eliminatedusing
§
new degrees of freedom, often as additional scalar, vec- the relation:
tor or tensor fields. It is not immediately obvious that
8πG a2δρ=E U (48)
the behaviour of these theories can be be encapsulated 0 ∆− ∆
by a either a (Q,ηslip) or (g˜,ζ) parameterization. There E can be expressed in terms of Φ,Φ˙ and Ψ in the con-
∆
is a riskthatwe mightdevelopa model-independentfor-
formalNewtoniangauge,seeeqns.(18). Rearrangingand
malism that does not map onto most of our well-studied
redefining the coefficient functions, we obtain:
theories.
In VI we will study several example cases, chosen to U∆ =k2α~T0χ~ +kα~T1χ~˙ +A0k2Φ + A1kΦ˙
§
be representative of common classes of modified gravity + F k2Ψ+F kΨ˙ (49)
0 1
models, and ask whether they can be expressed in the
This procedure has enabled us to eliminate energy
two-function format of quasistatic PPF. Attempting to
density fluctuations of ordinary matter from U . If the
map disparate theories onto a single framework is only ∆
new d.o.f. are not coupled to ordinary matter then δρ
plausible if those theories share some common features.
does not appear in U anyway. It is worth reminding
Hence, before turning to specific examples, we wish to ∆
the reader that in this section we are working in the
considerwhatgeneralstatementscanbemadeaboutthe
conformal Newtonian gauge, so Φ and Ψ should not
structureofthe the fieldequationsintheorieswithextra
be confused with their gauge-invariant counterparts Φˆ
degrees of freedom.
and Ψˆ, which already contain first- and second-order
Letthe scalarperturbationstoanextradegreeoffree-
derivatives respectively. If we were to consider eqn.(49)
dombedenotedbyχ,e.g. ifthenewdegreeoffreedomis
ascalarfieldφthenχ=δφ. (We havesuccumbedtothe in a general gauge we would find that Φˆ˙ and Ψˆ only
common but unwise choice of terminology by referring appear in the combination Γˆ. The Ψ˙ in eqn.(49) would
to ‘scalar perturbations’, even though the new degree of become Ξ˙ recall that in the conformal Newtonian
−
freedommayitselfbeavectorortensorfield. ‘Spin-0per- gauge Φˆ Φ 1χ and Ψˆ Ψ Ξ, using eqns.(15)
turbations’wouldbeabetterchoiceofterminology[38]). ≡ (cid:0)≡−6 ≡ ≡−
and (16) .
If we are to obtain second-order field equations then we
knowthatU∆andUΘ canonlycontaintheperturbations Weap(cid:1)plyasimilartreatmenttotheremainingcompo-
χ and χ˙.
nents of δU . Recall that U and U are permitted to
More generally we can introduce multiple new d.o.f. µν P Σ
containsecond-ordertermssuchasΦ¨ andχ¨. Termssuch
and denote their scalar perturbations by the vector χ~,
with components χ(i). The perturbed field equations in as θ˙,Π˙,Π¨,Σ˙ and Σ¨ are discarded to maintain contain
consistency with our treatment of U ; we stress again
a general gauge are awkward and rarely used; hence we ∆
that this is done only on an intuitive basis. The result-
will specialise to the conformal Newtonian gauge for the
ing expressions for δU (together with eqn.(49)) are as
remainder of this paper. The relevant expressions for µν
follows:
scalar-tensor theory ( VIA) are presented in a general
gauge in Appendix A.§ U = kβ~Tχ~ +β~Tχ~˙ +B kΦ+B Φ˙ +I kΨ+I Ψ˙ (50)
Θ 0 1 0 1 0 1
In the conformal Newtonian gauge the 00-element of U = k2~γTχ~ +k~γTχ~˙ +~γTχ~¨+C k2Φ+C kΦ˙
the tensor δU can be represented as: P 0 1 2 0 1
µν
+C Φ¨ +J k2Ψ+J kΨ˙ +J Ψ¨ (51)
2 0 1 2
U∆ = k2α~T0χ~ +kα~T1χ~˙ +A0k2Φ+A1kΦ˙ U = ~εTχ~ + 1~εTχ~˙ + 1 ~εTχ~¨+D Φ+ D1 Φ˙
+F k2Ψ+F kΨ˙ + M0δρ+ M1δρ˙ (47) Σ 0 k 1 k2 2 0 k
0 1 k2 k3 D K K
+ 2Φ¨ +K Ψ+ 1Ψ˙ + 2Ψ¨ (52)
where δρ is the total energy density fluctuation of stan- k2 0 k k2
dard cosmological fluids (similarly for θ, Π and Σ to be where β ,...ε and B ,...K denote functions of back-
i i i i
used shortly). α~ and α~ denote vectors of functions ground quantities.
0 1
10
The Bianchi identities then give us two constraint Colddarkmatter,radiation,massiveneutrinos,WIMPs,
equations coupling terms in χ(i),Φ,Ψ and their deriva- scalar fields and a cosmological constant can all be re-
tives. Incontrasttotheprevioussectionwecannolonger covered as limiting cases of GDM. In our case the d.o.f.
set the coefficients of the each term to zero individually. parameterizedasGDMmaybegenuinefluidcomponents
Inthe casewithoutextrafieldsthis waspossiblebecause (e.g. scalar or vector fields), or effective fluids (eg. the
all our variables were non-dynamical, so obtaining evo- scalaron of f(R) gravity, the Weyl fluid of DGP gravity
lution equationsfor them wouldbe unphysical. But now -see VI).Forexample,ineqn.(49)weidentifythe extra
§
that extra fields appear in δU , the Bianchi identities d.o.f. with an energy density perturbation:
µν
yield equations describing how the metric variables re-
spond to the set of perturbations χ~. Therefore we no α~T0χ~ˆ+α~T1χ~ˆ˙ =8πG0a2ρE∆E (53)
longer have a hierarchy of constraint equations for the
where the ‘hat’ symbol indicates that we have folded
coeffcients α ,...K that allow us to reduce them down
0 2 in the necessary metric perturbations to make gauge-
to two functions. This is not problematic in itself. To
invariant versions of χ and χ˙. Similarly we can con-
map a specific theory onto the parameterization we can
simplypullthenecessarycoefficientsoutoftheperturbed struct gauge-invariant χ~ˆ, χ~ˆ˙ and χ~¨ˆ from the terms in
field equations. We will see shortly ( VI) that in many eqns.(50)-(52), and identify these with velocity, pressure
cases that these are relatively simple f§unctions. andanisotropicstressperturbations respectively. Asub-
scriptE willbeusedtoindicatetheseeffectiveperturba-
To constrain a general parameterized theory using
tions.
MarkovChainMonteCarlo(MCMC)analysisweinstead
The GDM formalism then provides a way of reducing
choose some sensible ansatz for the functions α ,...K .
0 2
thesefourfluidperturbationstojusttwo,whichcanthen
Forexample,a Taylorseriesupto cubic orderinΩ was
Λ
berelatedtothemetricpotentialsviatheperturbedcon-
used in [39, 40]; the MCMC then constrains the coeffi-
servation equations. These are [41]:
cients of the terms in the Taylor series. Rigidly fixing
the format of the parameterization in this way means δ˙ = (1+w ) k2θ 3Φ˙ + w˙EδE 3 w Γ +
that we simply have to constrain real numbers. This E − E E − 1+w − H E E
E
simplicityisakeyadvantageofexplicitlyparameterizing [...]Φ+[...]Φ˙ +[(cid:16)...]Φ¨ +[...(cid:17)]Ψ+[...]Ψ˙ +[...]Ψ¨ (54)
for the new fields as in eqns.(49-52). The alternative ap-
proach – absorbing the new fields into an evolving Geff c2 δ +w Γ 2
and slip parameter where possible – will give Geff and ζ θ˙E = −H 1−3c2ad θE + ad(1E+w E) E − 3ΣE +
very complicated forms that are difficult to parameter- E
ize (for example, see eqns.(76) and (96)). The trade-off [...]Φ+[..(cid:0).]Φ˙ +[..(cid:1).]Φ¨ +[...]Ψ+[...]Ψ˙ +[...]Ψ¨ (55)
is that our method requires considerably more than two
where the square brackets denote combinations of the
coefficient functions. We expect that some of these will
functions B ,...K . We should remember that these are
i i
be well-constrained by the data, others less so. really second-order equations, due to the χ~ˆ˙ in eqn.(53).
In the case of just one or two new d.o.f., the system
In eqn.(55) the pressure perturbation of the effective
consistingoftheEinsteinequations,thetwoconservation
fluid has been decomposed into an adiabatic and a non-
equations for ordinarymatter andtwo Bianchiidentities
adiabatic part:
for the U-tensor can be solved. In order to avoid a con-
tradiction, the Bianchi identities for the U-tensor must Π =c2 δ +w Γ (56)
E ad E E E
be equivalent to the equations of motion for the extra
We have adopted common notation by using Γ to rep-
degrees of freedom (obtained by varying the action with E
resent the dimensionless non-adiabatic pressure pertur-
respectto the extra fields or similar). Futhermore,when
bation; this should not be confused with our metric po-
only a single d.o.f. is present the solutions of the two
tential Γˆ. The adiabatic sound speed is fully determined
components of the Bianchi identity must be consistent
by the equation of state parameter w :
with each other. E
Whenmorethantwonewd.o.f. arepresenttheBianchi c2 =w 1 w˙E (57)
identities do not provide sufficient information to solve ad E − 3 1+w
E
H
the system, and we must supply additional relations be-
The non-adiabatic pressure perturbation is specified by
tween the new d.o.f, metric variables and matter vari-
introducing a parameter c2 , interpreted as the sound
ables. With our goal of an abstract, unified framework eff
speed of the fluid in its rest frame:
in mind, we will introduce a general structure to tackle
suchcases. Wemakethe conjecturethatonecanusethe w Γ = c2 c2 δ +3 1+w θ (58)
E E eff − ad E H E E
‘generalizeddarkmatter’(GDM)formalismdevelopedby
A third (cid:0)and final(cid:1)p(cid:0)arameter(cid:0)is need(cid:1)ed (cid:1)to relate the
Hu[41]inordertoobtainthenecessaryclosurerelations.
anisotropic stress, Σ , to the velocity perturbations.
GDMisaphenomenologicalmodelinwhichspecification E
This is the viscosity parameter c2 :
of three parameters - an equation of state, a rest-frame vis
soundspeedandaviscoussoundspeed-sufficetorecon- w˙
(1+w ) Σ˙ +3 Σ EΣ =4c2 k2θ (59)
struct the full perturbed stress-energy tensor of a fluid. E E H E − w E vis E
E
(cid:16) (cid:17)