can mondian vector theories explain the cosmic speed up?

Can MONDian vector theories explain the cosmic speed up?

Vincenzo F. Cardone1,2,* and Ninfa Radicella2,3,†

1Dipartimento di Fisica Generale, ‘‘Amedeo Avogadro,’’ Via Pietro Giuria 1, 10125-Torino, Italy2INFN-Sezione di Torino, Via Pietro Giuria 1, 10125-Torino, Italy

3Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129-Torino, Italy(Received 20 May 2009; published 10 September 2009)

Generalized Einstein-Aether vector field models have been shown to provide, in the weak field regime,

modifications to gravity which can be reconciled with the successful modified Newtonian dynamics

(MOND) proposal. Very little is known, however, on the function F ðKÞ defining the vector field

Lagrangian so that an analysis of the viability of such theories at the cosmological scales has never

been performed. As a first step along this route, we rely on the relation between F ðKÞ and the MOND

interpolating function ða=a0Þ to assign the vector field Lagrangian thus obtaining what we refer to as

MONDian vector models. Since they are able by construction to recover the MOND successes on galaxy

scales, we investigate whether they can also drive the observed accelerated expansion by fitting the models

to the type Ia supernovae data. Should this be the case, we have a unified framework where both dark

energy and dark matter can be seen as different manifestations of a single vector field. It turns out that both

MONDian vector models are able to well fit the low redshift data on type Ia supernovae, while some

tension could be present in the high z regime.

DOI: 10.1103/PhysRevD.80.063515 PACS numbers: 98.80.k, 95.35.+d, 95.36.+x, 98.80.Es

I. INTRODUCTION

The old standard cosmological model of a universe fullof baryon matter only and regulated by the laws ofEinstein’s general relativity provided a remarkable pictureto consistently explain the background evolution of theUniverse from the initial singularity up to the presentday. Unfortunately, two observational facts then put thiselegant scenario in serious trouble. On the one hand, itturned out that the gravitational field measured on differentscales may not be reconciled with what is inferred frombaryonic matter only, the most famous example beingrepresented by the flat rotation curves of spiral galaxies[1]. On the other hand, recent data on the Hubble diagramof type Ia supernovae (hereafter SNeIa) provided the firstevidence for an accelerated expansion [2]. Surprising as itwas, the cosmic speedup has been then confirmed byupdated SNeIa data [3–7], the anisotropy spectrum of thecosmic microwave background radiation (CMBR) as mea-sured by both balloon [8] and satellite [9,10] experiments,and the data on the large scale clustering of galaxies [11] asestimated from large spectroscopic galaxy surveys.

Both these problems have been traditionally solved byadding new ingredients to the cosmic pie, while leavingunchanged the underlying theory. Cold dark matter (CDM)has been then invoked to fill the gap between the baryonmass (constrained by the big bang nucleosynthesis) andwhat is needed to reproduce the observed gravitationalfield. Moreover, this mysterious component must not in-teract with anything except gravitationally so as to escapedetection, notwithstanding the incredible efforts spent up

to now to find direct evidence for any dark particle. At thecosmological scales, dark matter still behaves as normalmatter so that a further component is needed in order todrive the accelerated expansion. This new actor on thescene must, moreover, have an unusual negative pressureand not cluster on galactic and cluster scales in order to notoppose gravitational clustering. Referred to as dark energy,the nature and nurture of this new term is still debated withthe cosmological constant [12] being the simplest can-didate. Although added by hand ad hoc, the cosmologicalmodel comprising both of them and known as CDM isable to fit extremely well the full data set at available , thusearning the name of concordance model. Notwithstandingthis remarkable success, the CDM is theoretically unap-pealing because of several well-known shortcomingswhich motivated the search for other dark energy candi-dates such as quintessence scalar fields [14] and modifiedgravity theories, either introducing higher dimensions [15]or correcting the gravitational Lagrangian [16,17].It is worth noting that both dark matter and dark energy

can be avoided if one accepts that the problem is not withwhat is missing, but rather with how we describe gravity.From this point of view, all the evidences for dark matterand dark energy should be rather seen as evidence thatsomething is wrong with the Newton-Einstein theory ofgravity. As a next step, one has therefore to look for a wayto modify gravity at both the galactic and cosmologicalscales possibly finding a unified mechanism explainingphenomena taking place on very different scales such asthe flatness of rotation curves and the acceleration of theUniverse expansion.Generalized Einstein-Aether theories are one of these

proposed mechanisms based on the introduction of dy-namical timelike vector field with noncanonical kinetic

*[email protected]†[email protected]

PHYSICAL REVIEW D 80, 063515 (2009)

1550-7998=2009=80(6)=063515(15) 063515-1 2009 The American Physical Society

http://dx.doi.org/10.1103/PhysRevD.80.063515

terms. Such a proposal is not fully new, but builds upon theextensive analysis of standard Einstein-Aether theories[18] as phenomenological probes of Lorenz violation inquantum gravity. Indeed, the nonvanishing expectationvalue of the Aether field will dynamically select a preferredframe in the spacetime, namely, the one in which the timecoordinate basis vector @t is aligned with the direction ofthe vector field, thus leading to violation of the Lorenz andgauge invariance. The interest in these models was thenrenewed after the demonstration [19] that tensor-vector-scalar theory (TeVeS), i.e. the fully relativistic and com-plete modified Newtonian dynamics (MOND) theory [20],can be reformulated as a generalized Einstein-Aether the-ory. Later, it has been shown that it is possible to get aMOND-like behavior in the weak field regime of vectormodels [21]. Motivated by these considerations, we ex-plore here the cosmological viability of these theories bysuitably setting the vector field Lagrangian in such a waythat the successful MOND phenomenology is recovered inthe Newtonian limit.

The structure of the paper is as follows. The basicequations and properties of the vector models are describedin Sec. II where we also explain how we choose the vectorfield Lagrangian. Section III is devoted to a detailed de-scription of the statistical methodology and the data used totest the models in the low redshift regime with a discussionof the results obtained from the likelihood analysis. Asimilar test is then presented and discussed in Sec. IVwhere we add high redshift data to probe this complemen-tary redshift range. Conclusions and perspectives are fi-nally given in Sec. V.

II. THE MODEL

A general action for a vector field A coupled to gravitycan be written as

S ¼Z

d4xffiffiffiffiffiffiffig

p R

16GN

þLðg; AÞþ SM; (1)

where GN is the Newton gravitational constant, g themetric (with the signature ;þ;þ;þ), R the Ricci scalar,and SM the matter action. The vector field Lagrangian Lmay be any covariant and local function but, following[21], we will only consider the case

L ðg; AÞ ¼ M2

16GN

F ðKÞ þ ðAA þ 1Þ16GN

; (2)

where F is a generic function of

K ¼ M2KrA

rA (3)

K ¼ c1g

g þ c2

þ c3

: (4)

It is worth noticing that K represents all the possiblecanonical kinetic terms that we introduce in the vectorLagrangian through the generic function F ðKÞ, thus al-

lowing one to have also a noncanonical contribution bythese terms. The ci quantities are dimensionless constants,while M is a scaling mass parameter, and a nondynam-ical Lagrange multiplier with dimensions of mass squared.It is possible to show [21–23] that the Einstein field

equations may still be formally written as

G ¼ ~T þ 8GNTM (5)

with G and TM the usual Einstein and matter stress-

energy tensors, while ~T contains the terms related to the

vector field and some of its derivatives (see, e.g., [21] for itsfull expression). The equation of motion for the vector fieldreads

rðF 0JÞ þF 0y ¼ 2A (6)

with F 0 ¼ dF =dK and

J ¼ ðK þK

ÞrA (7)

and the functional derivative

y ¼ rArA

ðK

ÞA

: (8)

Introducing the flat Robertson-Walker metric in Eq. (5)then gives the modified Friedmann equations which read

1 K1=2 d

dK

F

K1=2

H2 ¼ 8GN

3; (9)

d

dt

dFdK

2

H

¼ 8GNðþ pÞ; (10)

where, because of the metric symmetry, it is

K ¼ 3H2

M2¼ 3

"2H2

H20

: (11)

Here, we have defined two new parameters, namely " ¼M=H0 and ¼ c1 þ 3c2 þ c3, while ð; pÞ are the totalenergy density and pressure of the source terms (matter,radiation, neutrinos, etc.) andH ¼ _a=a is the usual Hubbleparameter. Hereafter, we will denote derivatives with re-spect to t with a dot and to K with a prime, and use asubscript 0 to label present-day quantities.It is convenient to rearrange the above equations in a

different form. To this end, we first solve Eq. (9) to get

F 0ðKÞ ¼ 1 ðzÞcritE

2ðzÞ þF ðKÞ2K

(12)

with EðzÞ ¼ HðzÞ=H0, the dimensionless Hubble parame-ter. Inserting Eq. (12) into Eq. (10) and using

d

dt¼ ð1þ zÞH0EðzÞ ddz

to change the variable, one finally gets

VINCENZO F. CARDONE AND NINFA RADICELLA PHYSICAL REVIEW D 80, 063515 (2009)

063515-2

d

dz

"2F ðKÞ6E2ðzÞ =crit

E2ðzÞ 1

E2ðzÞ

¼ 3ðþ pÞ=crit

ð1þ zÞEðzÞ(13)

with crit ¼ 3H20=8GN , the present-day critical density.

Let us now assume that the Universe is filled by a matter-like term and radiation. Since the vector field is onlycoupled to gravity, the continuity equation

_þ 3Hðþ pÞ ¼ 0

still holds for both matter and radiation and we can there-fore write the right-hand side of Eq. (13) as

3ðþ pÞ=crit

ð1þ zÞEðzÞ ¼ ½3M þ 4rð1þ zÞð1þ zÞ2EðzÞ (14)

with i ¼ iðz ¼ 0Þ=crit, the present-day density pa-rameter of the i-th component.

Equations (12) and (13) clearly show that a key role indetermining the cosmic evolution is played by the func-tional expression adopted forF ðKÞ. A guide to the choiceof this quantity is provided by the observation that, in thenonrelativistic regime, the field equations reduce to

r

2þ c1dFdK

r

¼ 8GN (15)

with

K ¼ c1jrj2M2

; (16)

being the gravitational potential sourced by the densitydistribution . By using the above expression, we get

r ½ðjrj=MÞr ¼ 8GN; (17)

which is the modified Poisson equation for the MONDtheory,1 provided we identify the MOND interpolatingfunction ða=a0Þ with

ða=a0Þ ¼ ðffiffiffiffiffiffiK

pÞ ¼ 2þ c1

dFdK

: (18)

Note that, by this position, the MOND acceleration scalea0 turns out to be related to the mass scale M as

a0 ¼ "cH0ffiffiffiffiffiffiffiffiffic1p , c1 ¼

"cH0

a0

2; (19)

where we have reintroduced the speed of light c. It is worthnoting that such a model thus provides a natural mecha-nism to explain why one observationally finds that a0 cH0 which now emerges as a consequence of the local and

cosmological phenomena being different manifestations ofthe same underlying theory.Since we know that MOND makes it possible to fit the

flat rotation curves of spiral galaxies (see, e.g., [25] andreferences therein), it is reasonable to assume that a viableexpression forF ðKÞ should lead to the same interpolatingfunction ða=a0Þ which is successfully used on localscales. Two such functions are the simple form [26]

ðxÞ ¼ x

1þ x; (20)

and the standard form [24]

ðxÞ ¼ xffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2

p : (21)

Although Eq. (21) has been the first proposal to be testedwith success [25], recent analyses [26,27] seem to favorEq. (20). However, in order to gain further insight on theproblem of which interpolating function is better moti-vated, we will consider both cases, contrasting themagainst data probing radically different scales.Inserting alternatively Eqs. (20) and (21) into Eq. (18)

gives, respectively,

F ðKÞ ¼ 4

c1½

ffiffiffiffiffiffiK

p lnð1þ

ffiffiffiffiffiffiK

pÞ; (22)

F ðKÞ ¼ 2

c1½K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Kð1þKÞp þ lnðffiffiffiffiffiffiK

pþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þK

pÞ:

(23)

Hereafter, we will refer to the models assigned by Eqs. (22)and (23) as the simple and standard MONDian vectormodel in order to clearly remember that the main ingre-dient of this kind of cosmological theory is the presence ofa vector field with a Lagrangian constructed in such a wayto recover the simple or standard MOND interpolating

function. Both expressions for F ðKÞ depend onffiffiffiffiffiffiK

pso

that it must be > 0 in the cosmological setting. In thesmall K limit, F ðKÞ K for both the simple and stan-dard MONDian vector models so that we can use theperturbative analysis in [28] for the Einstein-Aether mod-els to see whether the condition > 0 is compatible withthe constraints on the coefficients ci. The requirement thatthe Hamiltonian for the perturbations is positive definitelyimplies c1 < 0, while the constraint ðc1 þ c2 þ c3Þ=c1 0has to be set in order to avoid tachyonic propagation ofspin-0 modes. On the other hand, if we allow superluminalpropagations of both spin-0 and spin-2 modes, as supportedin [29], we then get that, for c2 > 0 and

ðc1 þ 3c2Þ c3 ðc1 þ c2Þ;it is indeed > 0 so that we can safely consider theMONDian vector models without violating any constrainton the ci coefficients.

1Actually, such a modified Poisson equation was proposed forMOND in the framework of the so called AQUAL theory [24],one of the first attempts to work out a relativistic MOND theory.AQUAL was later abandoned since it turned out to be unable toaccount for lensing data without cold dark matter thus being incontrast with the original MOND philosophy.

CAN MONDian VECTOR THEORIES EXPLAIN THE . . . PHYSICAL REVIEW D 80, 063515 (2009)

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A. The simple MONDian vector model

In order to determine the dynamics for this case, we haveto insert Eq. (22) into the master equation (13) and solvewith respect to EðzÞ. Somewhat surprisingly, after somealgebra, we get

Q ½z; EðzÞ dE

dz¼ 0

with Q½z; EðzÞ an algebraic function of redshift z, thedimensionless Hubble parameter EðzÞ, and the constantparameter of the model. Since we know that the Universeis expanding, dE=dz > 0 so that we must solveQ½z; EðzÞ ¼ 0. Rearranging the different terms gives

y3 þ y2 f½1 lnð1þ yÞ þ ½M þrð1þ zÞ ð1þ zÞ3gyþ lnð1þ yÞ

¼ ½M þrð1þ zÞð1þ zÞ3g; (24)

where we have set y ¼ EðzÞ and defined

¼ffiffiffiffiffiffi3

p"

; (25)

¼ 2"2

3c1¼ 2a20

3c2H20

’ 0:01h2 (26)

with h ¼ H0=ð100=km=s=MpcÞ. Note that, in the right-hand side of Eq. (26), we have used Eq. (19) and set a0 ¼1:2 1010 m=s2 in agreement with the estimates comingfrom the MOND fit to the galaxy rotation curves.

In order to reduce the number of parameters of themodel, we can insert Eq. (22) into (12) and evaluate it atz ¼ 0. Remembering that Eðz ¼ 0Þ ¼ 1 by definition, wethen get

lnð1þ Þ

1þ ¼ 1M r

; (27)

which can be solved numerically for given values ofðM;r; hÞ. Actually, since r is typically set by theCMB temperature, the simple MONDian vector model isfully characterized by only two parameters. This is thesame as the concordance CDM model where the sameparameters ðM; hÞ have to be assigned in order to com-pare the theory with the data. However, in that scenario,two different ingredients are invoked in order to explain thedynamics of galaxies and the cosmic speedup, while herethe solution to both problems comes out as a consequenceof the presence of a single vector field.

B. The standard MONDian vector model

We can repeat the same steps as before to get the masterequation for the case whenF ðKÞ is given by Eq. (23). Notsurprisingly given the similarities of the models, we stillget an algebraic relation

2þ 2 þ

yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2y2

p 3yffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2y2

p

¼ 2½M þrð1þ zÞð1þ zÞ3 þ lnðyþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2y2

p Þy2

(28)

with y, , and defined as above. The relation between and the other model parameters may be obtained as beforeand turns out to be

lnðþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2

qÞ ð1 2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 2p ¼ 2ð1M rÞ

;

(29)

which has still to be solved numerically. Note that, as forthe simple case, also the standard MONDian vector modelis characterized by only two quantities, namely, the totalmatter density parameter M and the present-day scaledHubble constant h.

III. TESTING THE LOW REDSHIFT REGIME

Any model that aims to describe the evolution of theUniverse must be able to reproduce what is indeed ob-served. This is particularly true for the models we areconsidering, since we are introducing a single vector fieldto get rid of both dark matter and dark energy. Matching themodel with observations is also a powerful tool to constrainits parameters and, as an interesting by-product, allows usto estimate some quantities common to every model (suchas the age of the Universe t0 and the transition redshift zT)to previous literature values.As a first test, we are here interested in exploring the

behavior of our models in the low redshift regime so thatwe start by only considering data probing up to z 1:5. Tothis end, we therefore maximize the following likelihoodfunction:

LðpÞ / LSNeIaðpÞ exp 1

2

!obs

b !thb

!b

2

exp

1

2

hHST h

h

2; (30)

where p denotes the set of model parameters. Beforediscussing in detail the term related to the SNeIa data,we concentrate on the two Gaussian priors. The first onetakes into account the constraints on the physical baryondensity !b ¼ bh

2 with

!obsb !b ¼ 0:0228 0:0055

as estimated in [30] and in agreement with what is inferredfrom the abundance of light elements [31]. The HST KeyProject [32] has estimated the Hubble constant H0 using awell-calibrated set of local distance scale estimators thusending up with


063515-4

hHST h ¼ 0:720 0:008

as a final model-independent constraint.

A. The SNeIa data

Since the first announcement [2] of the evidence ofcosmic speedup, the importance of the SNeIa Hubblediagram as a probe of the Universe background evolutionhas always been clear. It has therefore become a sort ofground zero for every proposed cosmological model to fitthe SNeIa data. To this end, one relies on the predicteddistance modulus2

thðz;pÞ ¼ 25þ 5 log

c

H0

ð1þ zÞrðz;pÞ

(31)

with rðzÞ the dimensionless comoving distance

rðz;pÞ ¼Z z

0

dz0

Eðz0;pÞ : (32)

The likelihood function is then defined as

L SNeIaðpÞ ¼ 1

ð2ÞNSNeIa=2jC1SNeIaj1=2

exp

C1

SNeIa T

2

; (33)

where NSNeIa is the total number of SNeIa used, is a NSNeIa-dimensional vector with the values ofobsðziÞ thðziÞ, and CSNeIa is the NSNeIa NSNeIa co-variance matrix of the SNeIa data. Note that, if we neglectthe correlation induced by systematic errors, CSNeIa is adiagonal matrix so that Eq. (33) simplifies to

L SNeIaðpÞ / exp½ 2SNeIaðpÞ=2 (34)

with

2SNeIaðpÞ ¼

XNSNeIa

i¼1

obsðziÞ thðziÞ

i

2

(35)

withi the error on the observed distance modulusobsðziÞfor the i-th object at redshift zi. As input data, we use theUnion SNeIa sample assembled in [33] by reanalyzingwith the same pipeline both the recent SNeIa SNLS [5]and ESSENCE [6] samples and older nearby and highredshift [4] data sets.

B. How many model parameters ?

As shown by Eqs. (24) and (28), in order to determinethe cosmic dynamics, one has to set the value of thepresent-day total matter density parameter M. On theother hand, the Gaussian prior on !b in the likelihoodfunction (30) and of the distance priors introduced laterasks for discriminating between the baryons only and total

matter physical densities. It is therefore worth wonderinghow M and b are related. Since our theory reduces toMOND in the low energy limit and MOND does not needany cold dark matter on the galactic scales (hence no othermatter than the visible one), we could argue thatM ¼ b

should hold. On the other hand, large amounts of missingmatter are needed in order for MOND to reproduce theobserved phenomenology on cluster scales [34]. It wastherefore postulated [35] that massive neutrinos (withm 2 eV) can play the role of dark mass in galaxyclusters. Moreover, solar and atmospheric neutrino experi-ments [36] have shown that the three active neutrinos fromthe standard model of particle physics mix their flavorswhich is only possible if they are massive. Finally, it is alsoworth noting that it has been claimed [37] that a singlesterile neutrino with mass in the range 4–6 eV is bettersuited to explain the results of the MiniBooNE experiment,while a 11 eV sterile neutrino has been indeed advocated[38] in order to solve the MOND problems on cluster andcosmological scales.Massive neutrinos decoupled at 1 MeV and since last

scattering they have been nonrelativistic particles so that,from a cosmological point of view, they behave exactly asmatter. Assuming three families of degenerate neutrinos,the total matter density parameter will therefore read

M ¼ b þ 3m

94h2 eV; (36)

so that the number of model parameters is increased by oneupdating from ðM; hÞ to ðb;m; hÞ.It is worth stressing, however, that discriminating be-

tween M as a single quantity and M as function ofðb;m; hÞ is only possible if the data at hand depend onthem separately. To understand this point, let us considerthe SNeIa Hubble diagram. In order to fit this data set, wejust need the dimensionless Hubble parameter EðzÞ whichis obtained by solving, e.g., Eq. (24). To this end, we justhave to set the value ofM so that all the SNeIa likelihoodwill be a function of ðM; hÞ only and we cannot set anyconstraint on ðb; mÞ. That is why we have added theGaussian priors on !b and h in order to make the like-lihood explicitly dependent on b. However, since thelikelihood is mainly driven by the SNeIa term, one canforecast that the constraints on ðb;mÞ will be quite weakbecause of the degeneracy being only partially broken.

C. Results

In order to maximize the likelihood function (30), werun a Markov chain Monte Carlo (MCMC) algorithm toefficiently explore the parameter3 space. We use a singlechain with 100 000 points which reduces to 3100 after

2We use lnx and logx to denote the logarithm base e and 10.

3Note that, hereafter, we use logm rather than m as neutrinomass parameter since this choice allows one to explore a widerrange.


063515-5

we cut out the burn in period and thin the chain. Thehistograms of the values of each parameter are then usedto infer the median and the mean and the 68 and 95%confidence ranges summarized in Table I. The best-fitparameters (i.e. the set maximizing the likelihood) forthe simple MONDian vector model turn out to be

ðb; logm; hÞ ¼ ð0:056; 0:54; 0:70Þ;while the standard case gives

ðb; logm; hÞ ¼ ð0:050; 0:55; 0:70Þ:For both models we get 2

SNeIa=DOF ’ 1:02, where DOF is

degrees of freedom, thus indicating a very good agreement.Moreover, the physical baryon density reads !b ¼ 0:0276(0.0246) for the simple (standard) model in good agree-ment with the observed value well within the 1 error,while both models predict values for h very close to theHST Key Project result. It is worth noting that a baryon-only Universe may be safely excluded since the reduced 2

values are of order 2.15 for both cases thus definitivelyruling out models with no massive neutrinos.

A look at Table I shows that the constraints on both b

and logm are quite weak. Moreover, the best-fit values ofboth parameters radically differ from their median values.This is, however, not an unexpected result. As we havesaid, should the prior on!b be neglected, the model shouldcollapse into a two-parameter one with M replacing theset ðb; logmÞ. The prior on !b thus helps in discrimi-nating between the two but, from the point of view ofmaximizing the likelihood, it is of little help. Indeed, theonly reliable constraint is on M reading

hMi ¼ 0:28; M;med ¼ 0:28;

68%CL: ð0:25; 0:31Þ;95%CL: ð0:23; 0:34Þ;

for the simple MONDian vector model and

hMi ¼ 0:28; M;med ¼ 0:28;

68%CL: ð0:25; 0:31Þ;95%CL: ð0:22; 0:33Þ;

for the standard case. Note that these values are in verygood agreement with typical estimates from previousanalyses of comparable data sets [7,10,33]. It is also worthnoting that the results are almost fully independent on thefunctional expression adopted for F ðKÞ which is an ex-pected consequence of the two models matching each otherin order to fit the same SNeIa Hubble diagram. Providedthat the above constraints on M are met, we can chooseany value ofb (for a given h) and then find a correspond-ing logm value giving rise to a model with a given like-lihood value, i.e. L depends only on ðM; hÞ. As aconsequence, it is therefore not surprising that the con-straints on ðb; logmÞ are so weak.With this caveat in mind, it is nevertheless interesting to

look at the neutrino mass. Converting the constraints onlogm into constraints on m (in eV), we get

hmi ¼ 2:3; m;med ¼ 2:5;

68% CL: ð0:7; 3:6Þ;95% CL: ð0:1; 4:2Þ;


hmi ¼ 2:0; m;med ¼ 2:2;

68% CL: ð0:1; 3:4Þ;95% CL: ð0:0; 4:4Þ;

for the standard case. As already stated, atmospheric andsolar neutrino experiments have shown that the three fam-ilies of standard model neutrinos are massive, but they areunable to put any constraints on their exact masses, beingonly sensitive to mass squared differences. An upper limiton the mass may instead be set from the study of the tritium decay. By this method, the Mainz-Troitz experiment [39]was able to find m 2:2 eV. The median m valuesquoted above are smaller than this upper limit, while the68 and 95% confidence ranges do indeed suggest that it ispossible to fit the data equally well with still lighter neu-trinos. On the other hand, it is worth stressing that suchestimates rely on our assumption that three degeneratemassive neutrinos are present so that their total densityparameter reads ¼ 3m=94h

2. It has, however, been

TABLE I. Summary of the results of the likelihood analysis including SNeIa and Gaussianpriors on !b and h. The upper and lower parts of the table refer to the simple and standardMONDian vector model, respectively.

Par x hxi xmed 68% CL 95% CL

b 0.13 0.12 (0.05, 0.23) (0.02, 0.29)

logm 0.24 0.40 ð0:16; 0:55Þ ð0:88; 0:63Þh 0.700 0.700 (0.694, 0.706) (0.689, 0.711)

b 0.15 0.13 (0.07, 0.26) (0.02, 0.31)

logm 0:09 0.35 ð1:10; 0:53Þ ð2:40; 0:64Þh 0.700 0.700 (0.694, 0.706) (0.688, 0.712)


063515-6

claimed [37] that a single sterile neutrino with mass in therange 4–6 eV is better suited to explain the results of theMiniBooNE experiment. Should this be the case, the con-straints above should be multiplied by three thus giving asingle sterile neutrino with mass in the 95% confidencerange (0.3,12.6) eV. It is worth noting that an 11 eV sterileneutrino has been indeed advocated [38] in order to solvethe MOND problems on cluster and cosmological scales. Itis therefore tempting to investigate whether the results in[38] still hold for our vector models since on cluster scalesboth theories reduce to MOND.

The MCMC algorithm also makes it possible to inferconstraints on some interesting derived quantities. To thisend, one just has to evaluate a given function fðpÞ for eachpoint of the chain and then use the values thus obtained aswas already done for the parameters p. Table II summa-rizes the results for the present-day deceleration parameterq0, the transition redshift zT obtained by solving qðzTÞ ¼0, and the age of the Universe t0 estimated as

t0 ¼ tHZ 1

0

dz

ð1þ zÞEðzÞ (37)

with tH ¼ 9:78h1 Gyr, the Hubble time.As a first issue, let us consider the value of q0. Its

estimate is typically model dependent since it comes outas a derived quantity given a model parametrization. Inorder to escape this problem, one may resort to cosmo-graphic analyses based only on Taylor expanding the scalefactor. Using this approach, Cattoen and Visser [40] havefound values between q0 ¼ 0:48 0:17 and q0 ¼0:75 0:17 depending on the details of the methodused to fit the SNLS data set. A similar analysis but usingthe gamma-ray bursts as distance indicators allowedCapozziello and Izzo [41] to find values between q0 ¼0:94 0:30 and q0 ¼ 0:39 0:11 still in accordancewith our estimates. A different approach has been insteadadopted by Elgarøy and Multamaki [42], advocating amodel-independent parametrization of qðzÞ. Dependingon the SNeIa sample used and the parametrization adopted,their best-fit values for q0 range between0:29 and1:1,in good agreement with the estimates in Table II.

On the contrary, there is some conflict with the previousestimates of the transition redshift zT . For instance, using

the Gold SNeIa sample and linearly expanding qðzÞ, Riesset al. [4] found zT ¼ 0:46 0:13 in agreement with theupdated result zT ¼ 0:49þ0:14

0:07 obtained by Cunha [43] us-

ing the Union sample. Although there is a possible mar-ginal agreement within the 95% confidence range, wenevertheless consider this unsatisfactory result not a seri-ous flaw of our models since the estimate of zT is stronglymodel dependent so that it is not possible to decide whetherthe disagreement is with the data or with the fiducial modelused to fit the data.Finally, we note that the age of the Universe is in

agreement with previous estimates in the literature.Fitting the WMAP5 data and the SNLS SNeIa samplewith a prior on the acoustic peak parameter gives [10] t0 ¼13:73 0:12 Gyr in almost perfect agreement with theresults in Table II. Moreover, the shape of the color-magnitude diagram of globular clusters provides amodel-independent estimate, namely t0 ¼ 12:6þ3:4

2:6 Gyr[44], still in considerably good agreement with our values.Summarizing, the very good fits to the SNeIa data and

the agreement between observed and predicted derivedquantities make us confident that both the simple andstandard MONDian vector models successfully reproducethe data, thus being viable alternatives to the usual darkenergy models in the low redshift regime.

IV. PROBING THE HIGH REDSHIFT REGIME

The SNeIa Hubble diagram and the Gaussian priors on!b and h allow us to test the behavior of the MONDianvector models only over the redshift range probed by thesedata. Considering that the farthest SN has redshift z ’ 1:6,it is worth wondering whether the models work well forhigher z. To answer this question, we change the likelihoodfunction adding a further term

L ðpÞ / LlowðpÞ LdpðpÞ (38)

with LlowðpÞ the likelihood term related to low redshiftdata given by Eq. (30), whileLdpðpÞ is the distance priors-dependent term which we detail below.

TABLE II. Constraints on derived quantities (t0 in Gyr) from the chains obtained fitting theSNeIa with Gaussian priors on !b and h. The upper and lower parts of the table refer to thesimple and standard MONDian vector model, respectively.


q0 0:57 0:58 ð0:62;0:53Þ ð0:65;0:49ÞzT 0.73 0.72 (0.65, 0.80) (0.58, 0.88)

t0 13.71 13.70 (13.43, 14.00) (13.18, 14.31)

q0 0:58 0:58 ð0:62;0:53Þ ð0:65;0:49ÞzT 0.73 0.72 (0.64, 0.81) (0.57, 0.88)

t0 13.72 13.71 (13.42, 14.02) (13.16, 14.29)


063515-7

A. The distance priors

While SNeIa probe only the background evolution of theUniverse over a limited redshift range (up to z ’ 1:5), theCMBR anisotropy spectrum and the matter power spec-trum measured by galaxy surveys make it possible to bothconstrain the dynamics of perturbations and test the modelup to the last scattering surface. However, such a programis both theoretically difficult and computationally demand-ing. On the one hand, we must develop the full theory ofperturbations to compute the Cl coefficients of the multi-pole expansion of the CMBR anisotropies and the matterpower spectrum PðkÞ. While this is already done for thestandard dark energy models, a full perturbation theory forthe vector models we are considering is still to be devel-oped. On the other hand, even for popular dark energymodels, computing both the Cl and PðkÞ represents a bottle-neck for the algorithms matching data with theories.Fortunately, many features of the CMB and matter powerspectra may be expressed as a function of a limited subsetof quantities which are instead easy and straightforward tocompute. Motivated by this consideration, it has becomepopular to summarize the main constraints coming fromCMBR and matter power spectra in what are defined asdistance priors.4

In the analysis of the WMAP5 data [10], Komatsu et al.demonstrated that most of the information in the WMAPpower spectrum may be summarized in a set of constraintson the following quantities:

(i) the physical baryon density

!b ¼ bh2 (39)

with b the density parameter of baryons only;(ii) the redshift zLS to the last scattering surface that we

approximate as [45]

zLS ¼ 1048ð1þ 0:001 24!0:738b Þ

½1þ g1ð!bÞ!g2ð!bÞM ; (40)

g1ð!bÞ ¼ 0:0738!0:238b

1þ 39:5!0:763b

; (41)

g2ð!bÞ ¼ 0:560

1þ 21:1!1:81b

; (42)

having denoted with !M ¼ Mh2 the total matter

physical density;

(iii) the acoustic scale [46]

lA ¼ ðc=H0ÞrðzLSÞrsðzLSÞ (43)

with rsðzLSÞ the size of the sound horizon at thedecoupling epoch given by

rsðaLSÞ ¼ c=H0ffiffiffi3

pZ aLS

0

da

a2EðaÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ Ra

p ; (44)

with aLS ¼ ð1þ zLSÞ1 and R ¼ 3b=4r;(iv) the shift parameter defined as [46]

R ¼ ffiffiffiffiffiffiffiffiM

prðzLSÞ: (45)

While this set of constraints relies on the CMBR data, noneof them has to do with the matter power spectrum. Toovercome this problem, Eisenstein et al. [47] introducedthe acoustic peak parameter defined as

A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiMH

20

qczm

DVðzmÞ; (46)

where zm is the median redshift of the galaxy survey usedto extract the matter power spectrum and the volumedistance is given by

DVðzÞ ¼

cz

HðzÞ cr2ðzÞH0

1=3

: (47)

The analysis of the correlation function of a sample ofmore than 46 000 luminous red galaxies from the SloanDigital Sky Survey (SDSS) allowed Eiseinstein et al. toestimate

A ¼ 0:469 0:017

thus offering another valuable constraint on the modelparameters.5 However, a more recent analysis of the sev-enth SDSS data release allowed Sanchez et al. [30] tomeasure the correlation function along both the radialand tangential directions thus allowing a more detailedtreatment. Adding the CMBR data in a joint analysis, theauthors then provided a new set of distance priors adding tothe four quantities quoted above a new one defined as

GðzmÞ ¼ rðzMÞ ½EðzmÞ0:8: (48)

In order to take into account the distance priors, we in-troduce the following likelihood function:

L dpðpÞ ¼ 1

ð2Þ5=2jCdpj1=2 exp

dpC

1dp

Tdp

2

; (49)

where dp is a five-dimensional vector whose i-th element4Actually, not all these quantities are indeed distances.

Nevertheless, the set of constraints is collectively referred towith this name since most of them are easily related to adistance. 5Note that A does not depend on h.


063515-8

is given bydp;i ¼ xi;obs xi;th with the subscripts obs and

th denoting the observed and theoretically predicted val-ues. The label i runs from 1 to 5 referring, respectively, to!b, zLS, lA, R, and GðzmÞ. Finally, we follow appendix Bof [30] to set the observed values and the covariance matrixof the distance priors.

B. Fitting SNeIa and distance priors

We now run our MCMC algorithm to explore the mod-els’ parameter spaces, maximizing the likelihood (38) andtaking care of both the SNeIa and distance priors data sets.The median and mean values and 68 and 95% confidenceranges are summarized in Table III for both the simple andstandard MONDian vector model. The best-fit parameters,i.e. the values of ðb; logm; hÞ that maximize the like-lihood, are not forced to be equal to the median valuesbecause of correlations among the parameters. Indeed, wefind

ðb; logm; hÞ ¼ ð0:0392;0:038; 0:753Þ

for the simple MONDian vector model, while it is

ðb; logm; hÞ ¼ ð0:0387;0:001; 0:753Þ

for the standard case. Both models, however, do not offer agood performance in fitting the data. Indeed, for the best-fitparameters, we get

2SNeIa=DOF ¼ 1:32; !b ¼ 0:0222; zLS ¼ 1081:7;

lA ¼ 300:0; R ¼ 1:52; GðzmÞ ¼ 1424;

for the simple case and

2SNeIa=DOF ¼ 1:31; !b ¼ 0:0219; zLS ¼ 1082:3;

lA ¼ 300:4; R ¼ 1:54; GðzmÞ ¼ 1424;

for the standard model. It is immediately clear from theunusually high reduced 2

SNeIa value signals that something

is going wrong with fitting the SNeIa Hubble diagram.Indeed, with DOF ¼ NSNeIa 3 ¼ 304, a reduced 2SNeIa=DOF ’ 1:3 has just a tiny 3 105 probability

to occur so that we can safely conclude that both modelsare not correctly fitting the SNeIa Hubble diagram data.This conclusion is further enforced comparing the abovebest-fit distance priors whose median values and standarddeviation6 are as follows:

!b ¼ 0:0228 0:0055; zLS ¼ 1090:1 0:9;

lA ¼ 301:6 0:7; R ¼ 1:701 0:018;

GðzmÞ ¼ 1175 21:

While the values of ð!b; zLS; lAÞ are in reasonable agree-ment, there are strong discrepancies for both the shift Rand the GðzmÞ parameters. Considering the 68 and 95%confidence ranges in Table IV does not ameliorate thecomparison so that we must conclude that the model isunable to fit both the SNeIa and distance priors data set.It is worth investigating why this happens. Actually, a

hint is given by noticing that the most discrepant parame-ters are those involving EðzÞ both directly, as for GðzmÞ, orindirectly through an integral, as both R and 2

SNeIa.

Taking the CDM model as a comparison, we indeedfind that both the simple and standard MONDian vectormodels systematically underestimate EðzÞ. As a conse-quence, the dimensionless comoving distance rðzÞ turnsout to be overestimated thus leading to ðzÞ becomingincreasingly higher than the concordance model predic-tion, as the redshift increases in accordance with the resultthatðzÞ is larger thanobs for SNeIa with z > 1. This alsoexplains why GðzmÞ gets larger than observed, while thesituation is different with the shift parameter. Indeed,according to Eq. (45), a larger rðzÞ should lead to a largerR, while we observe the opposite result. This can, how-ever, be easily explained considering that R is also pro-portional to M and we get

TABLE III. Summary of the results of the likelihood analysis. The upper and lower parts of thetable refer to the simple and standard MONDian vector model, respectively.


b 0.0394 0.0394 (0.0385, 0.0404) (0.0374, 0.0413)

logm 0:036 0:037 ð0:048;0:025Þ ð0:059;0:012Þh 0.753 0.753 (0.749, 0.757) (0.745, 0.761)

b 0.0384 0.0384 (0.0372, 0.0394) (0.0363, 0.0402)

logm 0.005 0.006 ð0:016; 0:025Þ ð0:031; 0:041Þh 0.753 0.753 (0.749, 0.758) (0.745, 0.761)

6Actually, the standard deviation is not a good estimator of theuncertainty on the distance priors because it does not take intoaccount the correlations among them. However, it gives an ideaof the discrepancy between predicted and observed values.


063515-9

hMi ¼ 0:091; M;med ¼ 0:091;

68% CL: ð0:089; 0:094Þ;95% CL: ð0:086; 0:096Þ;


hMi ¼ 0:095; M;med ¼ 0:095;

68% CL: ð0:094; 0:097Þ;95% CL: ð0:092; 0:098Þ;

for the standard case. These values are much smaller thanthe typical M ’ 0:26 [7,10,33] values obtained in litera-ture thus explaining why our shift parameter turns out to beso small notwithstanding the higher rðzLSÞ.

For completeness, we also summarize in Table V thevalues of the present-day deceleration parameter q0, thetransition redshift zT , and the age of the Universe t0. Theseresults strengthen our conclusion that both models, whileperforming well in the low redshift regime, are actuallyquite poor in reproducing data probing higher z. Indeed,the values of q0 are in reasonable agreement with theresults quoted in Sec. III C, even if our q0 values aresomewhat extreme. On the contrary, there is a clear dis-agreement with the previous estimates of the transitionredshift zT with our results being very high (also outsidethe range probed by the SNeIa Hubble diagram). Such aresult further signals that indeed the expansion rate and

hence the transition from acceleration to deceleration ofour models is too slow. Another evidence in favor of thisinterpretation is provided by t0 which turns out to be instrong disagreement with both the WMAP5 data andglobular clusters estimate. Indeed, Eq. (37) shows that,should we underestimate EðzÞ, the age of the Universeturns out to be higher which is just what happens forboth MONDian vector models.

C. A problem with the models or the data?

The large reduced 2SNeIa values, the disagreement be-

tween the best-fit predicted and observed distance priorsand the unacceptably high zT and t0 results should make usconclude that both MONDian vector models are unable tofit both the low and high redshift data. Comparing theresults in Table I and III, however, makes it evident thatthe two fitting procedures select very different regions ofthe parameter space. It is therefore worth wonderingwhether there is a problem with the distance priors datarather than with the models.To this end, it is worth stressing that, although widely

used in the recent literature (see, e.g., [7,10,33,48,49]),their estimate is actually model dependent. Indeed, in orderto obtain their central values and covariance matrix, onefirst fits a given model (typically the concordance CDMone) to the full CMBR anisotropy and galaxy power spec-tra data set using a Markov chain Monte Carlo method to

TABLE V. Constraints on derived quantities (with t0 in Gyr). The upper and lower parts of thetable refer to the simple and standard MONDian vector model, respectively.


q0 0:861 0:862 ð0:865;0:858Þ ð0:869;0:855ÞzT 1.71 1.71 (1.68, 1.73) (1.66, 1.76)

t0 16.93 16.93 (16.84, 17.01) (16.76, 17.10)

q0 0:856 0:856 ð0:859;0:854Þ ð0:861;0:852ÞzT 1.67 1.67 (1.65, 1.68) (1.64, 1.70)

t0 16.77 16.77 (16.66, 16.87) (16.60, 16.98)

TABLE IV. Constraints on the predicted distance priors parameters. The upper and lower partsof the table refer to the simple and standard MONDian vector model, respectively.


!b 0.0224 0.0223 (0.0219, 0.0229) (0.0213, 0.0234)

zLS 1081.6 1081.6 (1081.1, 1082.0) (1080.7, 1082.6)

lA 299.9 299.9 (299.2, 300.5) (298.6, 301.2)

R 1.524 1.523 (1.519, 1.529) (1.514, 1.533)

GðzmÞ 1424 1424 (1416, 1431) (1407, 1439)

!b 0.0218 0.0218 (0.0211, 0.0223) (0.0206, 0.0228)

zLS 1082.6 1082.5 (1081.8, 1083.4) (1081.2, 1084.0)

lA 300.6 300.6 (299.9, 301.4) (299.1, 302.2)

R 1.545 1.545 (1.543, 1.549) (1.539, 1.551)

GðzmÞ 1424 1424 (1416, 1432) (1409, 1439)


063515-10

sample the posterior probability. This same model is thenused to compute the distance priors along the chain andthen the sample thus obtained is analyzed to infer thecovariance matrix. As correctly stressed in [50], this pro-cedure relies on three main assumptions:

(1) the posterior probability from the CMBR anisotro-pies and galaxy power spectra is correctly describedby the distance priors, i.e. no information is lostwhen giving away the full data set in favor of thesimplified one;

(2) the mean and covariance matrix of the distancepriors parameters do not change when the modelspace is enlarged, i.e. the choice of the fiducialmodel does not affect the estimate of the priors;

(3) these summary parameters are weakly correlatedwith the input fiducial model parameters so thatthey can be used as independent constraints.

While the first and third assumptions have indeed beenverified [48,50], the second hypothesis cannot be fullyaddressed. On the one hand, one could argue that, unlessa significative amount of dark energy in the early Universeis present, all reliable models should match at high redshift,when the effect of dark energy fades away. As a conse-quence, using the CDM as a fiducial model to computequantities that mainly depend on the high redshift behaviorshould not affect the final estimate. However, even if thebackground evolution could be the same, it is still possiblethat the dynamics of perturbation is radically different as isthe case with modified gravity theories. This is indeed whatalso happens for the MONDian vector models we areconsidering. Fixing the model parameters to the best-fitvalues from the fit to the low redshift data only, we caneasily compare the luminosity distance with that for afiducial CDM scenario. As expected (since they bothreproduce the same data), the two can hardly be discrimi-nated over the range probed, while the Hubble parameteronly differs by a few percent. While the background evo-lution is therefore comparable, perturbations evolve in acompletely different way because of the presence of thevector field. As such, one cannot be sure that the informa-tion contained in the CMBR spectrum may still be sum-marized in the distance priors quantities as for the CDMmodel. A similar discussion also applies for the prior onGðzmÞ. Moreover, it is worth stressing that this latter quan-tity actually depends on data probing an intermediateredshift range so that one can not rely anymore on thefading of dark energy at high z as is typically done forquantities as, e.g., the acoustic scale lA and the shiftparameter R.

Investigating the impact of these problems on the esti-mate of the distance priors is outside our scope here. Wenevertheless stress that, because of the above considera-tions, one cannot safely reject the MONDian vector modelsbecause of the disagreement with the distance priors val-

ues. As a conservative conclusion, we are therefore forcedto only report the results being unable to decide whetherthe problem is with the data or with the fiducial model usedto retrieve them.

D. The effective EOS and the high z limit

An alternative way to compare the proposed MONDianvector models with standard dark energy models in boththe low and high redshift regime may be obtained byconsidering the effective equations of state (EOS).Indeed, from the point of view of the background evolu-tion, our models are equivalent to a cosmological scenariomade out of dust matter and a dark energy with an effectiveEOS given by

1þ weffðzÞ ¼2

3

d lnEðzÞd lnð1þ zÞ

Mð1þ zÞ3E2ðzÞ

1Mð1þ zÞ3

E2ðzÞ1

; (50)

so that the dark energy density parameter reads

DEðzÞ ¼ 1M

E2ðzÞ exp

3Z z

0

1þ weffðz0Þ1þ z0

dz0: (51)

Figures 1 and 2 show weffðzÞ and DEðzÞ for the standardand simple MONDian vector models setting the baryondensity parameter to the fiducial value b ¼ 0:04 andvarying the neutrino mass m. Note that the value of M

in Eqs. (50) and (51) is related to b and m throughEq. (36) so that changing m is the same as varying M.

0 0.5 1 1.5 2 2.5 3 3.5log 1 z

0.80.60.40.2

00.2

wef

f

0 2 4 6 8 10z

0.2

0.4

0.6

0.8

ΩD

E

FIG. 1. Effective EOS (left) and dark energy density parameter(right) for the standard MONDian vector model. We setðb; hÞ ¼ ð0:04; 0:70Þ and consider three values for logm,namely 1:0 (short dashed), 0.0 (solid), and 0.5 (long dashed).

0 0.5 1 1.5 2 2.5 3 3.5log 1 z

0.80.60.40.2

00.2

wef

f

0 2 4 6 8 10z

0.2

0.4

0.6

0.8

ΩD

E

FIG. 2. Same as Fig. 1, but for the simple MONDian vectormodel. The small ripples are due only to numerical errors.


063515-11

Both figures show that the value of m set the present-day EOS with weffðz ¼ 0Þ increasing as a function of m,even if there is a saturation at small m as can be easilyunderstood considering that the smaller is m, the less isthe impact of massive neutrinos in the energy budget. It isworth noting that for logm ’ 0:5, it is weffðz ¼ 0Þ ’ 1:0and dweff=dz takes very small values over the redshiftrange (0, 1), i.e. we recover (for both cases) a present-day cosmological constant. It is therefore not surprisingthat the fit to the SNeIa data set points toward such valuesof logm since these make both MONDian vector modelsas similar as possible to the CDM one over the rangeprobed by SNeIa themselves.

On the contrary, in the high redshift regime, the neutrinomass only plays a marginal role in determining the value ofweffðzÞ which stays almost constant at a value close to thedust one weff ¼ 0, i.e. the effective EOS approximatelytracks the matter term. However, the amount of dark en-ergy is different depending on m with lower values of theneutrino density giving rise to a largerDEðzÞ in the high zregime. Such a behavior is somewhat counterintuitive sinceone expects that the effective dark energy fades away in theearly Universe in order to recover the matter dominatedepoch. Actually, one must take into account that our matterterm is made out of baryons and massive neutrinos only.The lower is the neutrino mass, the higher must be thecontribution of the effective dark energy (that in this re-gime behaves as matter being weff ’ 0) to compensate forthe missing cold dark matter. Such a result can also beforecasted going back to Eq. (9) and noting that, for ourbest-fit models, we find 1 so that, for z 1, it isK 1 too. Let us then consider the simple MONDianvector model. Inserting Eq. (22) and taking the limit forEðzÞ 1, we approximately get

Mð1þ zÞ3 ’ E2 þ lnE ’ E2;

so that we indeed recover the usual expression for theHubble parameter for a matter dominated universe withM the total (baryons and massive neutrinos) matter term(having neglected radiation). From the point of view of theeffective dark energy formalism, if we reduce the neutri-nos’ contribution, we must add a further term acting asmatter and this is indeed provided by the effective darkenergy. This also explains whyDEðz 1Þ increases withdecreasing logm. A similar discussion also applies to thestandard MONDian vector models so that in both cases thefinal scenario is the one of a universe where the matter termdisappears to be replaced by a dark energy fluid actingapproximately as matter. This explains why the inclusionof the distance priors (probing the high z regime) pushesthe best fit toward smaller logm since, in this case, bothMONDian models recover the usual Friedmann modelswhich are known to successfully fit these high redshiftprobes. As already said, however, decreasing logm makes

weffðz ¼ 0Þ 1 thus worsening the fit to the SNeIaHubble diagram.

V. CONCLUSIONS

The astonishing successes of MOND on the galacticscales and the emergence of a relativistic theory playingthe role of its counterpart on cosmological scales haverenewed the interest in the search for a possible commonexplanation of both dark matter and dark energy phenome-nology. Vector theories are another way of recoveringMOND in the low energy limit so that it is worth wonder-ing whether they can also offer an elegant way of speedingup the cosmic evolution without the need of any darkenergy source. The ignorance about the form of the vectorfield Lagrangian may be bypassed relying on the linkbetween the function F ðKÞ and the MOND interpolatingfunction ða=a0Þ. Since dark matter is no more present,one should postulate the presence of massive neutrinos inorder to fill the gap between the total matter densityparameter and the baryon density alone. Moreover, suchmassive neutrinos are also advocated in order to reconcilethe results of solar and atmospheric neutrino experimentson the flavor mixing with the predictions of the standardmodel of particle physics.Motivated by these considerations, we have therefore

investigated the viability of two different MONDian vectormodels characterized byF ðKÞ expressions correspondingto the simple and standard MOND interpolating function.To this aim, we have first fitted them against the UnionSNeIa Hubble diagram using Gaussian priors on the physi-cal baryon density!b and the present-day Hubble constanth in order to break the ðb; logmÞ degeneracy. Bothmodels perform quite well, giving a perfect agreementwith the SNeIa data and previous estimates of the totalmatter density parameter M, the deceleration parameterq0, and the age of the Universe t0. Moreover, the (weak)constraints on the neutrino mass are consistent with theupper limits set by the Mainz-Troitz experiment assumingthree families of degenerate neutrinos. Should, instead, asingle sterile neutrino be the mass dominant component, itsestimated mass is in agreement with the constraints fromthe MiniBooNE experiment also falling in the right rangeadvocated to solve MOND problems on cluster scales.In order to investigate the high redshift behavior of the

models, we have repeated the likelihood analysis addingthe extended set of distance priors. It turns out that bothmodels should be rejected since they provide now a poor fitto the SNeIa data and strongly disagree with the observedshift R and GðzmÞ parameters. Distance priors are, how-ever, estimated through a model-dependent procedure sothat one cannot safely rely on them to exclude models thatare radically different from the fiducial one used to extractthe constraints onR and GðzmÞ. As a consequence, we areunable to conclude whether the disagreement is a failure ofthe MONDian vector models or an expected outcome of


063515-12

the different evolution of perturbations with respect tousual dark energy models because of the action of thevector field.

It is worth remembering that the search for a cosmologi-cal counterpart to MOND has a long history with vectormodels being only the most recent proposal. Applying aprocedure which is simply the generalization of the clas-sical derivation of the Friedmann equation from theNewtonian force law, but now starting from the MONDforce law, Lue and Starkman [51] derived an expression forthe Hubble parameter without referring to any underlyingmodified gravity theory. Defining gðxÞ ¼ H2=H2

0 and x ¼Mð1þ zÞ3, they indeed find

gðxÞ ¼ ð lnxþ c2Þx2=3 x < xcxþ c1x

2=3 x > xc(52)

with c1 ¼ c2 þ 3½lnð3Þ 1. Moreover, they also pro-posed a modified (by hand) version of this expression tobetter account for the low redshift behavior:

gðxÞ ¼8><>: x 0:1x2=3 lnð1þ zÞ 0:1 x ð3Þ3xþ 3½lnð3Þ 1x2=3 x > ð3Þ3:

(53)

Following [51], we set ð; xc;Þ ¼ ð15; 7 104; 0:7Þwith c2 ¼ 0 and show in Figs. 3 and 4 how Eqs. (52) and(53) compare to the standard and simple MONDian vectormodels. As it is better appreciated from the right panels,both Eqs. (52) and (53) agree reasonably well with the gðxÞexpression of our MONDian models for very high x, that isin the early Universe. This is an expected result since, inthis redshift range, both our models and the phenomeno-

logical Lue and Starkman proposals recover the typicalgeneral relativity matter-dominated scenario. On the con-trary, for small x, i.e. in the very low z regime, only themodified expression (53) matches reasonably well thosefor the standard and simple MONDian vector models as aconsequence of both mimicking an effective cosmologicalconstant. In the intermediate region, however, matching theLue and Starkman models to our own is impossible, sig-nalling that the strategy adopted by these authors is unableto trace the transition region to general relativity. However,it is also worth stressing that such a disagreement could beexpected since the Lue and Starkman procedure relies onthe assumption that, whatever the underlying modifiedgravity theory leading to (52) or (53) is, the Birkhofftheorem still holds. This is not the case for vector theories[52] so that the two approaches differ from the verybeginning.As already quoted in the introduction, much interest has

been devoted to the TeVeS theory as a relativistic MONDcounterpart. In particular, Skordis et al. [53] have alsoinvestigated the growth of structure in TeVeS also comput-ing the CMBR anisotropy spectrum. It turned out that theinclusion of a cosmological constant term and of massiveneutrinos with m ’ 2 eV may lead to a reasonably goodagreement with the data, although a detailed fitting has notbeen performed. Since both the vector and scalar fieldspresent in TeVeS do not contribute significantly to thedynamics, it is likely that this TeVeSþ model matcheswell both the SNeIa Hubble diagram and the distancepriors. However, comparing the Skordis et al. model toour MONDian vector theories is not possible given theradical differences between the two approaches. Indeed,TeVeS needs the scalar field to act as a dark-matterliketerm on galactic scales, while the vector field boosts thegrowth of perturbations during the driven backgroundexpansion. On the contrary, in our approach, the vectormodels modify the low energy limit Poisson equation, thusrecovering the MOND-like behavior, but also originatesthe cosmic speedup. In a sense, our approach is moreeconomical claiming the lowest possible number ofingredients.The positive results obtained in the low redshift regime

may be considered only as a ground-zero-level analysis.Indeed, fitting the SNeIa Hubble diagram only tells us thatthe two considered MONDian vector models predict thecorrect background evolution over the redshift rangeprobed by the data, i.e. up to z 1:5. Needless to say,more tests are needed in order to assess the viability ofthese models. On one hand, we can still investigate thebackground evolution by extending the redshift rangethrough the use of the gamma-ray bursts Hubble diagram[54]. On the other hand, a more significant and demandingtest should be fitting both the galaxy power spectrum andthe CMBR data set. However, both these tasks are quitedaunting from the theoretical point of view. Indeed, as far

0 2 4 6 8x

1

100

10000

1. 106

1. 108

gx

0 2 4 6 8x

400200

0200400600800

∆gg

%

FIG. 3. Left: the gðxÞ function for the best-fit standardMONDian vector model (solid line) compared to the Lue andStarkman proposals. Right: relative deviation of the proposalgðxÞ from that of our model. Short and long dashed lines refer togðxÞ given by Eqs. (52) and (53), respectively.

0 2 4 6 8x

1

100

10000

1. 106

1. 108

gx

0 2 4 6 8x

400200

0200400600800

∆gg

%

FIG. 4. Same as Fig. 3 but for the simple model.


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as we know, the CMBR spectrum has never been computedfor the class of vector theories we are considering so thatone should first write down the full set of perturbationequations to then modify a numerical code like CAMB[55]. Somewhat easier (even if still difficult) is to deal withthe problem of growth of structures in vector models.Indeed, such a study has already been performed in [56]where the authors considered a simple power-law choicefor F ðKÞ. Although a detailed fitting to the data was notperformed, these authors have convincingly shown that thenew degrees of freedom sourced by the vector field mayindeed boost the growth of structure even in absence of anydark matter. This result is a good starting point for testingour proposed MONDian vector models provided one ac-cordingly changes the F ðKÞ function and introducesmassive neutrinos into the game. Note that, since wewant to recover MOND on galactic scales, we are postu-lating no cold dark matter so that the observed galaxypower spectrum should be matched to the predicted onewith a bias parameter determined by the clustering prop-erties of the massive (sterile or not) neutrino. Therefore,such a test is particularly powerful and worth addressing asthe next step of our analysis.

It is worth noting that help investigating the viability ofour MONDian vector models may come from a nonastro-physical experiment. Indeed, one of the key ingredients in

both MONDian vector models is the presence of massiveneutrinos with m 2 eV. The KATRIN experiment [57]on the tritium decay should be able to constrain theelectron neutrino mass with a sensitivity of 0:2 eV.Should this experiment indeed find that neutrinos are lessmassive than, e.g., 1 eV, our model could be in serioustrouble unless one assumes that a single massive sterileneutrino does indeed exist.As a final remark, one could note that we have titled our

paper with a question so that the reader could now claim ananswer. Unfortunately, because of the uncertainties on theuse of the distance priors and the small redshift rangeprobed, the unique answer we can give after our analysisis only a (somewhat frustrating) maybe.

ACKNOWLEDGMENTS

It is a pleasure to thank A. Sanchez for making availablethe covariance matrix of the distance priors in electronicform and for the illuminating comments on their use. Wealso warmly thank G. Angus for the instructive discussionon sterile neutrinos in MOND, and A. Diaferio andA. Tartaglia for a careful reading of the manuscript.V. F. C. is supported by University of Torino and RegionePiemonte. Partial support from INFN Project PD51 isacknowledged too.

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can mondian vector theories explain the cosmic speed up?

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