text: E. Keiha¨nen et al.: Impact of beam deconvolution on noise properties As in Sect. , we compared the white noise covariance ma- χp2df Wn Ideal Real Ideal Real trix against white noise simulations, and full covariance matrix Full 0.9994 0.9996 0.9996 against ideal and realistic simulations. The χ2 results for full 30 1.0016 1.0016 four-year frequency maps and for 70 GHz subsets are shown in Figs. 23 and 24. 10 The mean reduced χ2 values for pixel covariance are in gen- eral closer to one than in the case of harmonic covariance. This 30 Yr1 0.9987 Reduced0 .χ92994 1.0005 is the eﬀect of regularization noise. We are adding to the maps a 1.0020 1.0179 noise component that is accurately modelled by the extra term in covariance matrix, thus improving the agreement with the matrix 10 and the maps. The χ2 mean is 1.01 for 30 GHz 4-year data, 1.0076 for 44 Yr2 1.0025 Reduced1 .χ02025 1.0034 30 GHz, and 1.0016 for 70 GHz. The latter is only slightly larger 1.0052 1.0210 than the one-sigma statistical variation. For 70 GHz subsets the 10 values range from 1.0017 to 1.0287. 6.4. Pre-whitened noise 30 Yr3 0.9986 Reduced0 .χ92985 0.9995 The power of the χ2 test is limited in the sense that it col- 1.0011 1.0167 lapses a complicated noise structure into a single number, which is not constrained from below. It is easy to conceive a situation 10 where two errors cancel to produce a χ2 result close to one. For instance, a simple overestimation of the noise rms in TOI domain Yr4 1.0012 Reduced1 .χ02018 1.0026 30 1.0045 1.0200 leads to an artiﬁcially low χ2 result. We complemented the validation tests by analysis of pre- 10 whitened noise. The types of tests we performed here require a real-valued data set. For this reason we can perform the test in 18/23 1.0015 Reduced1 χ.02022 1.0039 pixel space only. 30 1.0055 1.0287 Consider a pre-whitened noise map, computed as − 10 x = M 1m (44) 19/22 0.9986 Reduced0 .χ92986 0.9989 30 1.0003 1.0074 where m is the noise map, and M is the “square root” of the covariance matrix C, for which C = MMT. Matrix M is can be 10 constructed in multiple ways. We took it to be the Cholesky de- composition of C. If matrix C accurately describes the properties 30 20/21 0.9987 Reduced0 .χ92991 0.9998 of the noise map m, x wi(cid:104)ll be (cid:105)a vector of uncorrelated Gaussian 1.0017 1.0110 noise with unit variance, xxT = I. We build histograms of values of x and compare them to the- 10 oretical Gaussian distribution, in a fashion similar to the analysis presented in Stompor et al. (2002). We combine the 100 noise re- 1.0 1.06 1.0 1.06 1.0 1.06 alizations into the same plot to improve statistics. The variance Reduced χ2 of the distribution yields the usual χ2 test. Comparison of the full Fig. 24. Chi2 test in pixel space, for 70 GHz subsets. Left: white distribution against the expected Gaussian distribution provides a more stringent test. noise. Middle: ideal and realistic noise, independent destriping. We show the distributions for two selected simulation cases Right: ideal and realistic noise, full destriping. in Fig. 25, both with realistic noise. The ﬁrst example is one where the NCVM models the noise well, the 70 GHz frequency map. The second example is the worst case from earlier tests, the LFI18/23 horn pair map with full destriping. In both cases, noise before whitening is very non-uniform. We show for com- parison the Gaussian distribution with variance equal to that of the data. The variance is dominated by the high tails of the dis- tribution, leading to a broad distribution that does not ﬁt the data anywhere. On the right we show the pre-whitened noise, to- gether with the ideal Gaussian distribution with unit variance. In the case of 70 GHz frequency map, the distribution of whitened noise follows the Gaussian closely, indicating that the NCVM models the noise well. In the horn pair case there is a visible de- viation. There are no high tails in the distribution, the deviation manifests rather as a overall broadening of the distribution. In fact, the distribution matches very well a Gaussian shape with standard deviation of 1.015. To be more quantitative, we performed Kolmogorov- Smirnov test on the pre-whitened noise. We constructed the cu- 20 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: Cosmological disformal transformations to the Einstein frame ∗ and gravitational couplings with matter perturbations Shinji Tsujikawa1 (Dated: May 27, 2020) → The disformal transformation of metric g Ω2(φ)g + Γ(φ, X)∂ φ∂ φ, where φ is a scalar µν µν µ ν ﬁeld with the kinetic energy X = ∂ φ∂µφ/2, preserves the Lagrangian structure of Gleyzes-Langlois- µ Piazza-Vernizzi (GLPV) theories (which is the minimum extension of Horndeski theories). In the presence of matter, this transformation gives rise to a kinetic-type coupling between the scalar ﬁeld φ and matter. We consider the Einstein frame in which the second-order action of tensor perturbations on the isotropic cosmological background is of the same form as that in General Relativity and study the role of couplings at the levels of both background and linear perturbations. We show that the eﬀective gravitational potential felt by matter perturbations in the Einstein frame can be conveniently expressed in terms of the sum of a General Relativistic contribution and couplings induced by the modiﬁcation of gravity. For the theories in which the transformed action belongs to a class of Horndeski theories, there is no anisotropic stress between two gravitational potentials in the Einstein frame due to a gravitational de-mixing. We propose a concrete dark energy model encompassing Brans-Dicke theories as well as theories with the tensor propagation speed c diﬀerent t from 1. We clarify the correspondence between physical quantities in the Jordan/Einstein frames and study the evolution of gravitational potentials and matter perturbations from the matter-dominated epoch to today in both analytic and numerical approaches. I. INTRODUCTION action with the metric. This interaction can be clearly seen in the Einstein frame (EF) where the Lagrangian is described by the Einstein-Hilbert term plus a canonical The large-distance modiﬁcation of gravity has been un- scalar ﬁeld [24–26]. In the EF, matter ﬁelds feel a metric der active study in connection to the dark energy prob- g conformally related to the EF metric gˆ of the form lem [1, 2]. Modiﬁcations of the Einstein-Hilbert term µν µν gˆ = Ω2(φ)g , where Ω(φ) is a conformal factor that R/(16πG) in the Lagrangian of General Relativity (GR), µν µν depends on φ [4, 6, 27]. where R is the Ricci scalar and G is the Newton gravi- For the theories in which ﬁeld derivatives are cou- tational constant, generally give rise to a radiative scalar pled to the metric, the conformal transformation can degree of freedom φ [3, 4]. Provided that the ﬁfth force be extended to a more general mapping of the metric– mediated by this new degree of freedom is suppressed in dubbed the disformal transformation [28]. In fact, the the solar system through Vainshtein [5] or chameleon [6] structure of the Lagrangian in Horndeski theories is mechanisms, the same scalar ﬁeld can potentially be the preserved under the so-called disformal transformation source for the late-time cosmic acceleration. gˆ = Ω2(φ)g + Γ(φ)∂ φ∂ φ, where Ω and Γ are func- µν µν µ ν Horndeski theories [7] are known as the most general tions of φ [29, 30]. In the presence of matter ﬁelds, scalar-tensor theories with one scalar degree of freedom the disformal transformation helps us to understand the whose equations of motion are kept up to second-order physical content of mixing between φ and matter [31–38]. in time and spatial derivatives (see also Refs. [8]). Many Gleyzes, Langlois, Piazza, and Vernizzi (GLPV) [39] of dark energy models proposed in the literature– such proposed a generalized version of Horndeski theories by as f(R) gravity [9, 10], Brans-Dicke (BD) theory [11, 12], expressing the Horndeski Lagrangian in terms of the kinetic braidings [13], and Galileons [14, 15]– belong to Arnowitt-Deser-Misner (ADM) decomposition of space- a sub-class of Horndeski theories. In the presence of ad- time [40] with the choice of the unitary gauge. This ditional matter, the authors in Ref. [16] derived linear generally generates derivatives higher than second order, perturbation equations of motion on the ﬂat Friedmann- but there is only one radiative scalar degree of freedom on Lemaˆıtre-Robertson-Walker (FLRW) background to con- the FLRW background [41, 42]. The Lagrangian struc- front dark energy models in the framework of Horndeski ture of GLPV theories is preserved under the disformal theories with the observations of large-scale structures, transformation of the form weak lensing, and Cosmic Microwave Background (CMB) (see also Refs. [17–22]). gˆ = Ω2(φ)g + Γ(φ, X)∂ φ∂ φ , (1.1) µν µν µ ν In BD theories, a scalar degree of freedom φ is coupled to the Ricci scalar R of the form φR [23]. The frame in where Γ depends on φ and its kinetic energy X = which matter ﬁelds are minimally coupled to the metric is ∂ φ∂µφ/2 [43]. µ dubbed the Jordan frame (JF). The standard interpreta- In the single ﬁeld system, it was shown that the invari- tion of measurements is usually performed in this frame. ance of curvature perturbations ζ and tensor perturba- In the JF of BD theories, the scalar ﬁeld φ mediates a tions γ holds under the disformal transformation (1.1) ij ﬁfth force with matter through its gravitational inter- [44–46] (see also Refs. [35, 47–50] for related works). For

text: Cosmological

label: title

text: disformal

text: transformations

text: to

text: the

text: Einstein

text: frame

label: paragraph

text: and

text: gravitational

text: couplings

text: with

text: matter

text: perturbations

text: Shinji

label: author

text: (Dated:

text: May

label: date

label: abstract

text: The

text: disformal

text: transformation

text: of

text: metric

text: g

text: Γ(φ,

text: X)∂

text: φ∂

text: φ,

text: where

text: φ

text: is

text: a

text: scalar

text: µν

text: µν

text: µ

text: ν

text: ﬁeld

text: with

text: the

text: kinetic

text: energy

text: X

text: φ∂µφ/2,

text: preserves

text: the

text: Lagrangian

text: structure

text: of

text: Gleyzes-Langlois-

text: µ

text: Piazza-Vernizzi

text: (GLPV)

text: theories

text: (which

text: is

text: the

text: minimum

text: extension

text: of

text: Horndeski

text: theories).

text: In

text: the

text: presence

text: of

text: matter,

text: this

text: transformation

text: gives

text: rise

text: to

text: a

text: kinetic-type

text: coupling

text: between

text: the

text: scalar

text: ﬁeld

text: φ

text: and

text: matter.

text: We

text: consider

text: the

text: Einstein

text: frame

text: in

text: which

text: the

text: second-order

text: action

text: of

text: tensor

text: perturbations

text: on

text: the

text: isotropic

text: cosmological

text: background

text: is

text: of

text: the

text: same

text: form

text: as

text: that

text: in

text: General

text: Relativity

text: and

text: study

text: the

text: role

text: of

text: couplings

text: at

text: the

text: levels

text: of

text: both

text: background

text: and

text: linear

text: perturbations.

text: We

text: show

text: that

text: the

text: eﬀective

text: gravitational

text: potential

text: felt

text: by

text: matter

text: perturbations

text: in

text: the

text: Einstein

text: frame

text: can

text: be

text: conveniently

text: expressed

text: in

text: terms

text: of

text: the

text: sum

text: of

text: a

text: General

text: Relativistic

text: contribution

text: and

text: couplings

text: induced

text: by

text: the

text: modiﬁcation

text: of

text: gravity.

text: For

text: the

text: theories

text: in

text: which

text: the

text: transformed

text: action

text: belongs

text: to

text: a

text: class

text: of

text: Horndeski

text: theories,

text: there

text: is

text: no

text: anisotropic

text: stress

text: between

text: two

text: gravitational

text: potentials

text: in

text: the

text: Einstein

text: frame

text: due

text: to

text: a

text: gravitational

text: de-mixing.

text: We

text: propose

text: a

text: concrete

text: dark

text: energy

text: model

text: encompassing

text: Brans-Dicke

text: theories

text: as

text: well

text: as

text: theories

text: with

text: the

text: tensor

text: propagation

text: speed

text: c

text: diﬀerent

text: t

text: from

text: We

text: clarify

text: the

text: correspondence

text: between

text: physical

text: quantities

text: in

text: the

text: Jordan/Einstein

text: frames

text: and

text: study

text: the

text: evolution

text: of

text: gravitational

text: potentials

text: and

text: matter

text: perturbations

text: from

text: the

text: matter-dominated

text: epoch

text: to

text: today

text: in

text: both

text: analytic

text: and

text: numerical

text: approaches.

text: I.

text: INTRODUCTION

text: action

text: with

text: the

text: metric.

text: This

text: interaction

text: can

text: be

text: clearly

text: seen

text: in

text: the

text: Einstein

text: frame

text: (EF)

text: where

text: the

text: Lagrangian

text: is

text: described

text: by

text: the

text: Einstein-Hilbert

text: term

text: plus

text: a

text: canonical

text: The

text: large-distance

text: modiﬁcation

text: of

text: gravity

text: has

text: been

text: un-

text: scalar

text: ﬁeld

text: In

text: the

text: EF,

text: matter

text: ﬁelds

text: feel

text: a

text: metric

text: der

text: active

text: study

text: in

text: connection

text: to

text: the

text: dark

text: energy

text: prob-

text: g

text: conformally

text: related

text: to

text: the

text: EF

text: metric

text: gˆ

text: of

text: the

text: form

text: lem

text: Modiﬁcations

text: of

text: the

text: Einstein-Hilbert

text: term

text: µν

text: µν

text: gˆ

text: where

text: Ω(φ)

text: is

text: a

text: conformal

text: factor

text: that

text: in

text: the

text: Lagrangian

text: of

text: General

text: Relativity

text: (GR),

text: µν

text: µν

text: depends

text: on

text: φ

text: where

text: R

text: is

text: the

text: Ricci

text: scalar

text: and

text: G

text: is

text: the

text: Newton

text: gravi-

text: For

text: the

text: theories

text: in

text: which

text: ﬁeld

text: derivatives

text: are

text: cou-

text: tational

text: constant,

text: generally

text: give

text: rise

text: to

text: a

text: radiative

text: scalar

text: pled

text: to

text: the

text: metric,

text: the

text: conformal

text: transformation

text: can

text: degree

text: of

text: freedom

text: φ

text: Provided

text: that

text: the

text: ﬁfth

text: force

text: be

text: extended

text: to

text: a

text: more

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text: mapping

text: of

text: the

text: metric–

text: mediated

text: by

text: this

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text: degree

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text: freedom

text: is

text: suppressed

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text: dubbed

text: the

text: disformal

text: transformation

text: In

text: fact,

text: the

text: the

text: solar

text: system

text: through

text: Vainshtein

text: or

text: chameleon

text: structure

text: of

text: the

text: Lagrangian

text: in

text: Horndeski

text: theories

text: is

text: mechanisms,

text: the

text: same

text: scalar

text: ﬁeld

text: can

text: potentially

text: be

text: the

text: preserved

text: under

text: the

text: so-called

text: disformal

text: transformation

text: source

text: for

text: the

text: late-time

text: cosmic

text: acceleration.

text: gˆ

text: Γ(φ)∂

text: φ∂

text: φ,

text: where

text: Ω

text: and

text: Γ

text: are

text: func-

text: µν

text: µν

text: µ

text: ν

text: Horndeski

text: theories

text: are

text: known

text: as

text: the

text: most

text: general

text: tions

text: of

text: φ

text: In

text: the

text: presence

text: of

text: matter

text: ﬁelds,

text: scalar-tensor

text: theories

text: with

text: one

text: scalar

text: degree

text: of

text: freedom

text: the

text: disformal

text: transformation

text: helps

text: us

text: to

text: understand

text: the

text: whose

text: equations

text: of

text: motion

text: are

text: kept

text: up

text: to

text: second-order

text: physical

text: content

text: of

text: mixing

text: between

text: φ

text: and

text: matter

text: in

text: time

text: and

text: spatial

text: derivatives

text: (see

text: also

text: Refs.

text: Many

text: Gleyzes,

text: Langlois,

text: Piazza,

text: and

text: Vernizzi

text: (GLPV)

text: of

text: dark

text: energy

text: models

text: proposed

text: in

text: the

text: literature–

text: such

text: proposed

text: a

text: generalized

text: version

text: of

text: Horndeski

text: theories

text: by

text: as

text: f(R)

text: gravity

text: Brans-Dicke

text: (BD)

text: theory

text: expressing

text: the

text: Horndeski

text: Lagrangian

text: in

text: terms

text: of

text: the

text: kinetic

text: braidings

text: and

text: Galileons

text: belong

text: to

text: Arnowitt-Deser-Misner

text: (ADM)

text: decomposition

text: of

text: space-

text: a

text: sub-class

text: of

text: Horndeski

text: theories.

text: In

text: the

text: presence

text: of

text: ad-

text: time

text: with

text: the

text: choice

text: of

text: the

text: unitary

text: gauge.

text: This

text: ditional

text: matter,

text: the

text: authors

text: in

text: Ref.

text: derived

text: linear

text: generally

text: generates

text: derivatives

text: higher

text: than

text: second

text: order,

text: perturbation

text: equations

text: of

text: motion

text: on

text: the

text: ﬂat

text: Friedmann-

text: but

text: there

text: is

text: only

text: one

text: radiative

text: scalar

text: degree

text: of

text: freedom

text: on

text: Lemaˆıtre-Robertson-Walker

text: (FLRW)

text: background

text: to

text: con-

text: the

text: FLRW

text: background

text: The

text: Lagrangian

text: struc-

text: front

text: dark

text: energy

text: models

text: in

text: the

text: framework

text: of

text: Horndeski

text: ture

text: of

text: GLPV

text: theories

text: is

text: preserved

text: under

text: the

text: disformal

text: theories

text: with

text: the

text: observations

text: of

text: large-scale

text: structures,

text: transformation

text: of

text: the

text: form

text: weak

text: lensing,

text: and

text: Cosmic

text: Microwave

text: Background

text: (CMB)

text: (see

text: also

text: Refs.

text: gˆ

text: Γ(φ,

text: X)∂

text: φ∂

text: φ

text: µν

text: µν

text: µ

text: ν

text: In

text: BD

text: theories,

text: a

text: scalar

text: degree

text: of

text: freedom

text: φ

text: is

text: coupled

text: to

text: the

text: Ricci

text: scalar

text: R

text: of

text: the

text: form

text: φR

text: The

text: frame

text: in

text: where

text: Γ

text: depends

text: on

text: φ

text: and

text: its

text: kinetic

text: energy

text: X

text: which

text: matter

text: ﬁelds

text: are

text: minimally

text: coupled

text: to

text: the

text: metric

text: is

text: φ∂µφ/2

text: µ

text: dubbed

text: the

text: Jordan

text: frame

text: (JF).

text: The

text: standard

text: interpreta-

text: In

text: the

text: single

text: ﬁeld

text: system,

text: it

text: was

text: shown

text: that

text: the

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text: ance

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text: ζ

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text: JF

text: of

text: BD

text: theories,

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text: φ

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text: tions

text: γ

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text: disformal

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text: force

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text: For


text: 11 (cid:18) (cid:19) (cid:20) (cid:21) where Vˆ = V /c3 , Vˆ = V /c3 , and given by Eqs. (5.11)-(5.12), where 1 1 ti 2 2 ti − − − − 1 dφ 2 − 3 − µ = λ q 3q , µ = λ q 3q . (6.17) ρˆ = 1 (q q )2 + Vˆ (φ) 1 1 1 2 2 2 1 2 DE 2 dtˆ 2 1 2 (cid:18) −(cid:19) (cid:20) dφ (cid:21) +3(q q )M Hˆ , (6.21) For the theories with q1 = q2 (i.e., c2t = constant) the 1 2 pl dtˆ term Aˆ3 vanishes, in which case the Lagrangian (5.27) 1 dφ 2 − 3 − − corresponds to that of the canonical scalar ﬁeld φ coupled Pˆ = 1 (q q )2 Vˆ (φ) DE 2 dtˆ 2 1 2 to matter. − − d2φ B. Background dynamics (q q )M . (6.22) 1 2 pl dtˆ2 From Eqs. (4.7)-(4.9) the background equations of mo- The matter ﬂuid (cid:16)and the(cid:17)scalar ﬁeld φ obey Eqs. (4.35) tion in the JF are given by and (5.15), respectively, with − q + q − dφ − q q dφ (cid:15) − Q = 1 2 ρˆ 3Pˆ 1 2 ρˆ . (6.23) 3M2 H2F = φ˙2 + V 3M2 Hφ˙F + ρ, (6.18) 2M dtˆ M dtˆ pl 1 2 pl 1,φ pl pl − − 2M2 F H˙ = (cid:15)φ˙2 + M2 (F¨ HF˙ ) + ρ + P, (6.19) The ﬁrst term on the rhs of Eq. (6.23) arises for the pl 1 pl 1 1 ρ˙ + 3H(ρ + P) = 0 . (6.20) standard coupled dark energy scenario characterized by q = q and c2 = 1 [67]. The second term on the rhs of 1 2 ti (cid:54) Eq. (6.23) does not vanish for the theories with q = q 1 2 These equations depend on the function F1(φ) but not (i.e. time-varying c2). t on F2(φ), so the quantities q2 and c2ti are irrelevant to To discuss the background dynam(cid:112)ics in the EF, it is the background dynamics. This means that the theories convenient to introduce the dimensionless quantities with same q and diﬀerent q (i.e., same A and diﬀer- 1 2 4 ent B4) cannot be distinguished from each other for the ≡ √ 1 dφ ≡ √ Vˆ background cosmology in the JF [54]. xˆ1 6M Hˆ dtˆ , xˆ2 3M Hˆ , pl pl The disformal transformation to the EF corresponds ≡ ρˆ ≡ ρˆ to the change of tensor propagation speed squared c2 = Ωˆ DE , Ωˆ , −B /A to cˆ2 = Nˆ¯2, so the quantity q arises in the Et F. DE 3Mp2lHˆ 2 m 3Mp2lHˆ 2 4 4 t 2 (cid:48) Nevertheless we are dealing with the same physics in the ≡ Pˆ ≡ −M Vˆ ≡ −Hˆ w , µ pl ,φ , (cid:15)ˆ , (6.24) two frames, so any physical condition (such as the stabil- (cid:82) ρˆ Vˆ H Hˆ ity of ﬁxed points) should not be subject to change. In where a prime represents a derivative with respect to what follows we shall study the correspondence of back- N ˆ = Hˆdtˆ. Recall that the matter equation of state ground quantities between the EF and the JF. w is invariant under the disformal transformation to the 1. Einstein frame EF. In the following we assume that w is constant. From Eq. (5.11) there is the relation Ωˆ + Ωˆ = 1. DE m The background equations of motion in the EF are The variables xˆ and xˆ obey the diﬀerential equations 1 2 √ √ (cid:48) 6 − − { − − } { − − } − xˆ = [2q (3w 1) 2 6 1 + q (q q )(3w 1) xˆ + q 3(q q )2 2 (3w 1)xˆ2 1 4 1 1 1 2 1 1 1 2 1 { − } +2√3q + µ q (2 + 3w) xˆ2] + xˆ (cid:15)ˆ , (6.25) 2 1 2 1 H (cid:48) − 6 xˆ = µxˆ xˆ + xˆ (cid:15)ˆ , (6.26) 2 2 1 2 2 H √ 3 − − 3 − − 3 − { − − − } (cid:15)ˆ = (1 + w xˆ2) q (q q )(3w 1) + 6(q q ) 1 w + q (q q )(3w 1) xˆ H 2 2 2 1 1 2 2 1 2 1 1 2 1 −3{ − − }{ − − − } − 3 − − { − − } 3(q q )2 2 1 w + q (q q )(3w 1) xˆ2 [w (q q ) q (2 + 3w) 3q µ ]xˆ2 , (6.27) 4 1 2 1 1 2 1 2 1 2 1 2 2 − where (cid:15)ˆ is related to the eﬀective equation of state wˆ of the system, as wˆ = 1 + 2(cid:15)ˆ /3. The ﬁeld density H eﬀ eﬀ H ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: Vˆ

text: V

text: Vˆ

text: V

text: and

text: given

text: by

text: Eqs.

text: where

text: ti

text: ti

text: dφ

text: µ

text: λ

text: q

text: µ

text: λ

text: q

text: ρˆ

text: (q

text: q

text: Vˆ

text: (φ)

text: DE

text: dtˆ

text: dφ

text: q

text: )M

text: Hˆ

text: For

text: the

text: theories

text: with

text: (i.e.,

text: constant)

text: the

text: pl

text: dtˆ

text: term

text: vanishes,

text: in

text: which

text: case

text: the

text: Lagrangian

text: dφ

text: corresponds

text: to

text: that

text: of

text: the

text: canonical

text: scalar

text: ﬁeld

text: φ

text: coupled

text: Pˆ

text: (q

text: q

text: Vˆ

text: (φ)

text: DE

text: dtˆ

text: to

text: matter.

text: B.

text: Background

text: dynamics

text: (q

text: q

text: )M

text: pl

text: From

text: Eqs.

text: the

text: background

text: equations

text: of

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text: The

text: matter

text: ﬂuid

text: ﬁeld

text: φ

text: obey

text: Eqs.

text: tion

text: in

text: the

text: JF

text: are

text: given

text: by

text: and

text: respectively,

text: with

text: q

text: q

text: dφ

text: q

text: q

text: dφ

text: Q

text: ρˆ

text: ρˆ

text: φ˙2

text: V

text: Hφ˙F

text: ρ,

text: dtˆ

text: M

text: dtˆ

text: pl

text: pl

text: pl

text: pl

text: F

text: H˙

text: (F¨

text: HF˙

text: ρ

text: P,

text: The

text: ﬁrst

text: term

text: on

text: the

text: rhs

text: of

text: Eq.

text: arises

text: for

text: the

text: pl

text: pl

text: ρ˙

text: P)

text: standard

text: coupled

text: dark

text: energy

text: scenario

text: characterized

text: by

text: q

text: q

text: and

text: The

text: second

text: term

text: on

text: the

text: rhs

text: of

text: ti

text: Eq.

text: does

text: not

text: vanish

text: for

text: the

text: theories

text: with

text: q

text: q

text: These

text: equations

text: depend

text: on

text: the

text: function

text: but

text: not

text: (i.e.

text: time-varying

text: t

text: on

text: so

text: the

text: quantities

text: and

text: are

text: irrelevant

text: to

text: To

text: discuss

text: the

text: background

text: in

text: the

text: EF,

text: it

text: is

text: the

text: background

text: dynamics.

text: This

text: means

text: that

text: the

text: theories

text: convenient

text: to

text: introduce

text: the

text: dimensionless

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text: with

text: same

text: q

text: and

text: diﬀerent

text: q

text: (i.e.,

text: same

text: A

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text: cannot

text: be

text: distinguished

text: from

text: each

text: other

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text: the

text: dφ

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text: background

text: cosmology

text: in

text: the

text: JF

text: Hˆ

text: dtˆ

text: Hˆ

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text: pl

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text: disformal

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text: the

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text: ρˆ

text: ρˆ

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text: change

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text: tensor

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text: speed

text: squared

text: Ωˆ

text: DE

text: Ωˆ

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text: so

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text: the

text: Et

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text: DE

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text: we

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text: dealing

text: with

text: the

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text: Pˆ

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text: −Hˆ

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text: frames,

text: so

text: any

text: physical

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text: (such

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text: ρˆ

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text: H

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text: ﬁxed

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text: should

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text: be

text: subject

text: to

text: change.

text: In

text: where

text: a

text: prime

text: represents

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text: derivative

text: with

text: respect

text: to

text: what

text: follows

text: we

text: shall

text: study

text: the

text: correspondence

text: of

text: back-

text: N

text: ˆ

text: Hˆdtˆ.

text: Recall

text: that

text: the

text: matter

text: equation

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text: state

text: ground

text: quantities

text: between

text: the

text: EF

text: and

text: the

text: JF.

text: w

text: is

text: invariant

text: under

text: the

text: disformal

text: transformation

text: to

text: the

text: Einstein

text: frame

text: EF.

text: In

text: the

text: following

text: we

text: assume

text: that

text: w

text: is

text: constant.

text: From

text: Eq.

text: there

text: is

text: the

text: relation

text: Ωˆ

text: Ωˆ

text: DE

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text: background

text: equations

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text: motion

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text: the

text: EF

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text: 12 (cid:104) (cid:105) parameter and the equation of state are given, respectively, by √ − xˆ − { − − } Ωˆ = 1 Ωˆ = 1 2 6(q q ) + 2 3(q q )2 xˆ + xˆ2 , (6.28) DE m 2 1 2 √1 2 1 2 − − − − − (cid:48) − Pˆ 3[2 3(q q )2]xˆ2 6xˆ2 2 6(q√ q )(xˆ (cid:15)ˆ xˆ ) wˆ = DE = −1 2 − 1 2 1 2 − 1 H 1 . (6.29) DE ρˆDE 3[2 3(q1 q2)2]xˆ21 + 6xˆ22 + 6 6(q1 q2)xˆ1 If µ is constant, which is the case for the exponential rized in Table I. − potential Vˆ (φ) = V0e µφ/Mpl, there are ﬁve ﬁxed points For the ﬁxed point (a), the ﬁeld density parameter characterized by constant xˆ and xˆ . They are summa- reads 1 2 − { − − } − { − } q (3w 1)[q 4 3(q q )2 6q + 3 3q (q q )2 + 2q w] Ωˆ = 1 1 −1 2 − 2 −1 1 2 2 , (6.30) DE 3[1 w + q (q q )(3w 1)]2 1 1 2 so that both Ωˆ and xˆ vanish√for radiation (w = 1/3). (b1) and (b2) are 1, they are neither relevant to radia- DE 1 − − If q = q , it follows that xˆ = 6q (3w 1)/[3(1 w)], tion/matter eras nor the late-time cosmic acceleration. 2 1 − 1− 1 − (cid:112) wˆ = [2q2(1 3w)2 + 3w(1 w)]/[3(1 w)], and Ωˆ = The point (c) is the scalar-ﬁeld dominated point 2qe12ﬀ(3w − 11)2/[3(w − 1)2] for the point (a). When wD=E 0, (ΩˆDE = 1) re√levant to dar−k energy. When q2−= q1 we this corresponds to the φ-matter-dominated era (φMDE) have xˆ = µ/ 6, xˆ = 1 µ2/6, and wˆ = 1+µ2/3, 1 2 eﬀ (cid:54) [67(cid:54) ] characterized by wˆeﬀ = ΩˆDE = 2q12/3. Provided that so the cosmic a−cceleration occurs for µ2 < 2. For q2 = q1, w = 1/3, the point (a) is a kind of scaling solution with wˆeﬀ is close to 1 provided that q1, q2, µ are smaller than a constant ratio Ωˆ /Ωˆ . the order of 1. m DE The point (d) corresponds to the scaling solution with Since the eﬀective equations of state wˆ for the points the ﬁeld density parameter eﬀ − − { } − − q2(9w2 30w 23) + 2q [8µ + 3q 9 + w(4 + 3w) ] + 3(w + 1)[4 4q µ + 3q2(w 3)] Ωˆ = 1 1 −2 − 2 2 . (6.31) DE [2µ q + 3q 3w(q + q )]2 1 2 1 2 (cid:112) − − When q = q , we have that Ωˆ = [3(1 + w) q (3w of the point (d) are given by 2 − 1 − DE 1 1)(µ +−q1 3q1−w)]/(µ +−q1 3q1w)2 and wˆeﬀ = [µw + − ± − − q1(3w 1)]/[µ q1(3w 1)]. In this case the radiation κˆ(d) = −µ 3q1 + 3q2 −64 15(µ 3q1 + 3q2)2 . scaling solution corresponds to ΩˆDE = 4/µ2 and wˆeﬀ = 1,2 2(µ q1 + q2) 1/3, whereas the matter scaling solution is characterized (6.33) − (cid:38) (cid:28) by Ωˆ = (3+q µ+q2)/(µ+q )2 and wˆ = q /(µ+q ). For µ 10 and q , q 1, κˆ(d) are imaginary with neg- DE 1 1 1 eﬀ 1 1 1 2 1,2 If the ﬁxed point (d) is stable, the solutions approach it ative real parts, so the point (d) is a stable spiral. For during the radiation and matter eras. non-relativistic matter (w = 0), t(cid:112)he eigenvalues of the The stability of the above ﬁxed points can be analyzed point (d) read by considering small perturbations δxˆ and δxˆ about − ± 1 2 −3(µ q + 3q ) D e(6a.c2h6)ofcatnhebme.wTr(cid:18)ihtetelninien(cid:19)artihzeedfovr(cid:18)mersion(cid:19)of Eqs. (6.25) and κˆ(1d,2) = 2(21µ − q12+ 3q2) (d) , (6.34) − − − δxˆ(cid:48) M δxˆ where D(d) = 9−(µ q1 + 3q2)−2 24[3 4q12 +(cid:38)2q1(µ + 1(cid:48) = 1 , (6.32) 3q )][(µ+3q )2 5q (µ+3q ) 3+6q2]. For µ 10 and δxˆ δxˆ 2 (cid:28) 2 1 2 1 2 2 q , q 1, the eigenvalues (6.34) are again imaginary 1 2 M × with negative real parts. Thus, the ﬁrst potential on the wMhere is a 2 2 matrix. If the two eigenvalues κˆ1,2 of rhs of Eq. (6.16) leads to the scaling radiation and matter are negative (or imaginary with negative real parts), eras driven by the ﬁxed point (d) with µ = µ . 1 then the corresponding ﬁxed point is stable. The point (a) can be potentially relevant to radiation In the presence of radiation (w = 1/3), the eigenvalues and matter eras, but one of the eigenvalues is positive, ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: parameter

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text: the

text: equation

text: of

text: state

text: are

text: given,

text: respectively,

text: by

text: xˆ

text: Ωˆ

text: Ωˆ

text: q

text: q

text: xˆ

text: DE

text: m

text: √1

text: Pˆ

text: q

text: q

text: )(xˆ

text: xˆ

text: wˆ

text: DE

text: −1

text: H

text: DE

text: ρˆDE

text: If

text: µ

text: is

text: constant,

text: which

text: is

text: the

text: case

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text: the

text: exponential

text: rized

text: in

text: Table

text: I.

text: potential

text: Vˆ

text: (φ)

text: µφ/Mpl,

text: there

text: are

text: ﬁve

text: ﬁxed

text: points

text: For

text: the

text: ﬁxed

text: point

text: (a),

text: the

text: ﬁeld

text: density

text: parameter

text: characterized

text: by

text: constant

text: xˆ

text: and

text: xˆ

text: They

text: are

text: summa-

text: reads

text: q

text: q

text: (q

text: q

text: w]

text: Ωˆ

text: −1

text: −1

text: DE

text: w

text: q

text: (q

text: q

text: so

text: that

text: both

text: Ωˆ

text: and

text: xˆ

text: vanish√for

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text: (w

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text: they

text: are

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text: relevant

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text: q

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text: xˆ

text: w)],

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text: late-time

text: cosmic

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text: wˆ

text: w)],

text: and

text: Ωˆ

text: The

text: point

text: (c)

text: is

text: the

text: scalar-ﬁeld

text: dominated

text: point

text: for

text: the

text: point

text: (a).

text: When

text: wD=E

text: (ΩˆDE

text: re√levant

text: to

text: dar−k

text: energy.

text: When

text: we

text: this

text: corresponds

text: to

text: the

text: φ-matter-dominated

text: era

text: (φMDE)

text: have

text: xˆ

text: µ/

text: xˆ

text: and

text: wˆ

text: eﬀ

text: characterized

text: by

text: wˆeﬀ

text: ΩˆDE

text: Provided

text: that

text: so

text: the

text: cosmic

text: a−cceleration

text: occurs

text: for

text: µ2

text: For

text: w

text: the

text: point

text: (a)

text: is

text: a

text: kind

text: of

text: scaling

text: solution

text: with

text: wˆeﬀ

text: is

text: close

text: to

text: provided

text: that

text: µ

text: are

text: smaller

text: than

text: a

text: constant

text: ratio

text: Ωˆ

text: /Ωˆ

text: the

text: order

text: of

text: m

text: DE

text: The

text: point

text: (d)

text: corresponds

text: to

text: the

text: scaling

text: solution

text: with

text: Since

text: the

text: eﬀective

text: equations

text: of

text: state

text: wˆ

text: for

text: the

text: points

text: the

text: ﬁeld

text: density

text: parameter

text: eﬀ

text: µ

text: Ωˆ

text: −2

text: DE

text: q

text: q

text: When

text: q

text: q

text: we

text: have

text: that

text: Ωˆ

text: w)

text: q

text: of

text: the

text: point

text: (d)

text: are

text: given

text: by

text: DE

text: and

text: wˆeﬀ

text: [µw

text: In

text: this

text: case

text: the

text: radiation

text: κˆ(d)

text: −µ

text: scaling

text: solution

text: corresponds

text: to

text: ΩˆDE

text: and

text: wˆeﬀ

text: whereas

text: the

text: matter

text: scaling

text: solution

text: is

text: characterized

text: by

text: Ωˆ

text: and

text: wˆ

text: q

text: /(µ+q

text: For

text: µ

text: and

text: q

text: q

text: κˆ(d)

text: are

text: imaginary

text: with

text: neg-

text: DE

text: eﬀ

text: If

text: the

text: ﬁxed

text: point

text: (d)

text: is

text: stable,

text: the

text: solutions

text: approach

text: it

text: ative

text: real

text: parts,

text: so

text: the

text: point

text: (d)

text: is

text: a

text: stable

text: spiral.

text: For

text: during

text: the

text: radiation

text: and

text: matter

text: eras.

text: non-relativistic

text: matter

text: (w

text: eigenvalues

text: of

text: the

text: The

text: stability

text: of

text: the

text: above

text: ﬁxed

text: points

text: can

text: be

text: analyzed

text: point

text: (d)

text: read

text: by

text: considering

text: small

text: perturbations

text: δxˆ

text: and

text: δxˆ

text: about

text: −3(µ

text: q

text: D

text: Eqs.

text: and

text: (d)

text: M

text: δxˆ

text: where

text: D(d)

text: 9−(µ

text: For

text: µ

text: and

text: δxˆ

text: δxˆ

text: q

text: q

text: the

text: eigenvalues

text: are

text: again

text: imaginary

text: M

text: with

text: negative

text: real

text: parts.

text: Thus,

text: the

text: ﬁrst

text: potential

text: on

text: the

text: wMhere

text: is

text: a

text: matrix.

text: If

text: the

text: two

text: eigenvalues

text: of

text: rhs

text: of

text: Eq.

text: leads

text: to

text: the

text: scaling

text: radiation

text: and

text: matter

text: are

text: negative

text: (or

text: imaginary

text: with

text: negative

text: real

text: parts),

text: eras

text: driven

text: by

text: the

text: ﬁxed

text: point

text: (d)

text: with

text: µ

text: µ

text: then

text: the

text: corresponding

text: ﬁxed

text: point

text: is

text: stable.

text: The

text: point

text: (a)

text: can

text: be

text: potentially

text: relevant

text: to

text: radiation

text: In

text: the

text: presence

text: of

text: radiation

text: (w

text: the

text: eigenvalues

text: and

text: matter

text: eras,

text: but

text: one

text: of

text: the

text: eigenvalues

text: is

text: positive,

text: ##LTLine##

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text: 15 Eqs. (4.37)-(4.42) we obtain the perturbation equations Let us derive solutions along which ζˆ and χˆ stay nearly (cid:48) (cid:39) constant in the scaling matter era. Setting ζˆ 0 and ζˆ(cid:48) = δNˆNˆ¯ + 32ΩˆmVˆm , (6.59) χˆ(cid:48) (cid:39) 0 in E−qs. (6.5−9)√and (6.60), respectively, we obtain (cid:39) −3[1 c2(1 6qxˆ )]Ωˆ χˆ(cid:48) = −((cid:15)ˆH + 1)χ(cid:32)ˆ − ζˆ− δNˆNˆ¯(cid:33) , (6.60) χˆ 2Kˆ2[3ct2txˆ21 +− (1 − c12t)(cid:15)ˆHm]δˆm, √ (6.68) (cid:39) 3[(1 + (cid:15)ˆ )(1 c2) + c2xˆ (3xˆ + 6q)]Ωˆ δˆ(cid:48) = −3ζˆ(cid:16)(cid:48) − Kˆ2 χˆ − (cid:17)Vˆm , (6.61) ζˆ H2Kˆ2[3c2t√xˆ2 + (t1 1− c21)(cid:15)ˆ ] m δˆm, (6.69) c2 t 1 t H t − √ (cid:39) c2xˆ (3xˆ 6q(cid:15)ˆ ) Vˆm(cid:48) = − (cid:15)ˆH + 6qxˆ1 Vˆm − δNNˆ¯ˆ , (6.62) Vˆm(s) Kˆ2t[3c12txˆ21 +1 −(1√− c2tH)(cid:15)ˆH]δˆm, (6.70) − K δˆN (cid:39) − 3c2xˆ (3xˆ 6q(cid:15)ˆ )Ωˆ δNˆNˆ¯ = 3Ωˆmδˆm 6x2ˆ21ˆ2(χˆ + ζˆ) , (6.63) Nˆ¯ 2Kˆt2[13c2txˆ121 + (1 − cH2t)(cid:15)ˆHm]δˆm, (6.71) K ≡ N where, in the denominators of Eqs. (6.68)-(6.71), we where ˆ k/(aˆHˆ ). Taking the ˆ derivative of Eq. (6.62) K have neglected the terms without containing ˆ2. The and using other equati(cid:34)ons of motion, the ve(cid:35)locity poten- tial Vˆ obeys gravitational potentials Ψˆ and Φˆ, which are deﬁned by m Eq. (4.43), obey − (cid:48)(cid:48) (cid:34)C (cid:48) (1 c2)Ω(cid:35)ˆ K C Vˆ + Vˆ + (cid:16) t(cid:17) m ˆ2 + Vˆ − (cid:39) 3Ωˆ m 1 m√ 2c2txˆ21 2 m Φˆ = Ψˆ 2Kˆm2 δˆm , (6.72) − 6q K = χˆ + χˆ + ζˆ ˆ2 , (6.64) which is equivalent to Eq. (5.30) with Eq. (5.28). Since 3xˆ1 Aˆ = 0 for q = q , the condition 2HˆM2 /Nˆ¯ (cid:29) |Aˆ | 3 1 2 pl 3,Nˆ where used for the derivation of Eq. (5.30) is automatically sat- √ C − − isﬁed. 1 = [2(3xˆ1 6µ)√(xˆ21 1) Substituting Eqs. (6.(cid:20)68), (6.71), and (6.72) into Eq. (6.53), { − } +Ωˆ 3xˆ √2 6(µ + q) ]/(2xˆ ) , (6.65) we obtain m 1 1 C − − − = 3[12xˆ5 4 6µxˆ4 2xˆ3(9 + 2q2 + 3qµ 6Ωˆ ) (cid:39) − 3 (cid:21) 2 +√3xˆ {12qµ + Ωˆ 1(Ωˆ −11 − 2q(q + µ))} m Ψˆg 2Kˆ2 Ωˆmδˆ√m 1 + 2q2 √ 1 m m − − −√6xˆ21{4q − 4µ + (3q + 5µ)Ωˆm} +(1 c2t){( 6xˆ1 + 2q)−( 6xˆ1 } 2q(cid:15)ˆH) , (6.73) − − 2 3c2xˆ2 + (1 c2)(cid:15)ˆ 6(q + µ)Ωˆ (Ωˆ 1)]/(2xˆ ) . (6.66) √ t 1 t H m m 1 where xˆ = 6/[2(µ + q)] and (cid:15)ˆ = 3µ/[2(µ + q)] for the The general solution to Eq. (6.64) can be written in 1 H ﬁxed point (d) with w = 0. The eﬀective gravitational the form Vˆ = Vˆ (s) + Vˆ (h), where Vˆ (s) is the special m m m m coupling Gˆ during the scaling matter epoch reads solution and Vˆ (h) is the homogenous solution derived by eﬀ √ √ m sheotrtiiznogn tpheertruhrsbaotfioEnqs. s(a6t.6is4fy)intog bKˆe z(cid:29)ero1., Fthoer thhoemsougbe-- Gˆeﬀ (cid:39) 1 + 2q2 + (1 − c2t){( 6xˆ1 + 2q)−( 6xˆ1 −} 2q(cid:15)ˆH), G 2 3c2xˆ2 + (1 c2)(cid:15)ˆ nous solution induces the rapid oscillation of the veloc- t 1 t H (6.74) ity potential with a frequency associated with the term − K where the term 2q2 arises in BD theories after the con- (1 c2)Ωˆ ˆ2/(2c2xˆ2). t m t 1 formal transformation to the EF [67]. The last term on For the theories with q = 0, as long as the homogenous the rhs of Eq. (6.74) does not vanish for c2 diﬀerent from solution is initially suppressed relative to the special so- t 1. Since we are now considering the case where c2 is lution, it was found in Ref. [55] that the perturbations t constant, the variation of c2 does not appear in the ex- χ, ζ, and V in the JF stay nearly constant during the t m pression of Eq. (6.74). scaling matter epoch. As we conﬁrm later numerically, (cid:54) For the perturbations deep inside the Hubble radius, this is also the case for q = 0. At the scaling ﬁxed point the rhs of Eq. (5.19) can be neglected relative to the lhs (d) with w = 0 and µ = µ , the ratio Ωˆ /xˆ2 is constant. 1 m 1 of it. Then, during the scaling matter era, the matter We consider the sub-horizon perturbations satisfying the − K (cid:29) |C | perturbation obeys condition (1 (cid:34)c2t)Ωˆm ˆ2/(2c2txˆ21)(cid:35) 2 . Then, the spe- cial solution to Eq. (6.64(cid:16)) is giv(cid:17)en by (cid:48)(cid:48) µ − 2q (cid:48) − 3 Gˆ (cid:39) √ δˆ + δˆ eﬀ Ωˆ δˆ 0 . (6.75) m 2(µ + q) m 2 G m m (cid:39) − 6q 2c2xˆ2 Vˆ (s) χˆ + χˆ + ζˆ − t 1 . (6.67) (cid:29) m 3xˆ1 (1 c2t)Ωˆm Provided that µ q, there is a growing-mode solution ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Eqs.

text: we

text: obtain

text: the

text: perturbation

text: equations

text: Let

text: us

text: derive

text: solutions

text: along

text: which

text: ζˆ

text: and

text: χˆ

text: stay

text: nearly

text: constant

text: in

text: the

text: scaling

text: matter

text: era.

text: Setting

text: ζˆ

text: and

text: δNˆNˆ¯

text: in

text: E−qs.

text: respectively,

text: we

text: obtain

text: )]Ωˆ

text: ζˆ−

text: χˆ

text: χˆ

text: ζˆ

text: 1−

text: m

text: δˆm,

text: t

text: t

text: H

text: t

text: Vˆm

text: δNNˆ¯ˆ

text: Vˆm(s)

text: −(1√−

text: K

text: δˆN

text: )Ωˆ

text: δNˆNˆ¯

text: 3Ωˆmδˆm

text: ζˆ)

text: Nˆ¯

text: K

text: N

text: where,

text: in

text: the

text: denominators

text: of

text: Eqs.

text: we

text: where

text: ˆ

text: k/(aˆHˆ

text: Taking

text: the

text: ˆ

text: derivative

text: of

text: Eq.

text: K

text: have

text: neglected

text: the

text: terms

text: without

text: containing

text: The

text: and

text: using

text: other

text: of

text: motion,

text: the

text: poten-

text: tial

text: Vˆ

text: obeys

text: gravitational

text: potentials

text: Ψˆ

text: and

text: Φˆ,

text: which

text: are

text: deﬁned

text: by

text: m

text: Eq.

text: obey

text: K

text: C

text: Vˆ

text: Vˆ

text: m

text: ˆ2

text: Vˆ

text: 3Ωˆ

text: m

text: m√

text: m

text: Φˆ

text: Ψˆ

text: δˆm

text: K

text: χˆ

text: χˆ

text: ζˆ

text: ˆ2

text: which

text: is

text: equivalent

text: to

text: Eq.

text: with

text: Eq.

text: Since

text: Aˆ

text: for

text: q

text: q

text: the

text: condition

text: /Nˆ¯

text: |Aˆ

text: pl

text: where

text: used

text: for

text: the

text: derivation

text: of

text: Eq.

text: is

text: automatically

text: sat-

text: C

text: isﬁed.

text: Substituting

text: Eqs.

text: and

text: into

text: Eq.

text: +Ωˆ

text: √2

text: q)

text: we

text: obtain

text: m

text: C

text: 6Ωˆ

text: Ωˆ

text: 1(Ωˆ

text: µ))}

text: m

text: Ψˆg

text: Ωˆmδˆ√m

text: m

text: m

text: 4µ

text: µ)Ωˆ

text: (Ωˆ

text: t

text: t

text: H

text: m

text: m

text: where

text: xˆ

text: q)]

text: and

text: q)]

text: for

text: the

text: The

text: general

text: solution

text: to

text: Eq.

text: can

text: be

text: written

text: in

text: H

text: ﬁxed

text: point

text: (d)

text: with

text: w

text: The

text: eﬀective

text: gravitational

text: the

text: form

text: Vˆ

text: Vˆ

text: (s)

text: Vˆ

text: (h),

text: where

text: Vˆ

text: (s)

text: is

text: the

text: special

text: m

text: m

text: m

text: m

text: coupling

text: Gˆ

text: during

text: the

text: scaling

text: matter

text: epoch

text: reads

text: solution

text: and

text: Vˆ

text: (h)

text: is

text: the

text: homogenous

text: solution

text: derived

text: by

text: eﬀ

text: m

text: sheotrtiiznogn

text: tpheertruhrsbaotfioEnqs.

text: bKˆe

text: Fthoer

text: thhoemsougbe--

text: Gˆeﬀ

text: G

text: nous

text: solution

text: induces

text: the

text: rapid

text: oscillation

text: of

text: the

text: veloc-

text: t

text: t

text: H

text: ity

text: potential

text: with

text: a

text: frequency

text: associated

text: with

text: the

text: term

text: K

text: where

text: the

text: term

text: arises

text: in

text: BD

text: theories

text: after

text: the

text: con-

text: t

text: m

text: t

text: formal

text: transformation

text: to

text: the

text: EF

text: The

text: last

text: term

text: on

text: For

text: the

text: theories

text: with

text: q

text: as

text: long

text: as

text: the

text: homogenous

text: the

text: rhs

text: of

text: Eq.

text: does

text: not

text: vanish

text: for

text: diﬀerent

text: from

text: solution

text: is

text: initially

text: suppressed

text: relative

text: to

text: the

text: special

text: so-

text: t

text: Since

text: we

text: are

text: now

text: considering

text: the

text: case

text: where

text: is

text: lution,

text: it

text: was

text: found

text: in

text: Ref.

text: that

text: the

text: perturbations

text: t

text: constant,

text: the

text: variation

text: of

text: does

text: not

text: appear

text: in

text: the

text: ex-

text: χ,

text: ζ,

text: and

text: V

text: in

text: the

text: JF

text: stay

text: nearly

text: constant

text: during

text: the

text: t

text: m

text: pression

text: of

text: Eq.

text: scaling

text: matter

text: epoch.

text: As

text: we

text: conﬁrm

text: later

text: numerically,

text: For

text: the

text: perturbations

text: deep

text: inside

text: the

text: Hubble

text: radius,

text: this

text: is

text: also

text: the

text: case

text: for

text: q

text: At

text: the

text: scaling

text: ﬁxed

text: point

text: the

text: rhs

text: of

text: Eq.

text: can

text: be

text: neglected

text: relative

text: to

text: the

text: lhs

text: (d)

text: with

text: w

text: and

text: µ

text: µ

text: the

text: ratio

text: Ωˆ

text: is

text: constant.

text: m

text: of

text: it.

text: Then,

text: during

text: the

text: scaling

text: matter

text: era,

text: the

text: matter

text: We

text: consider

text: the

text: sub-horizon

text: perturbations

text: satisfying

text: the

text: K

text: |C

text: perturbation

text: obeys

text: condition

text: Then,

text: the

text: spe-

text: cial

text: solution

text: to

text: Eq.

text: is

text: by

text: µ

text: Gˆ

text: δˆ

text: δˆ

text: eﬀ

text: Ωˆ

text: δˆ

text: m

text: q)

text: m

text: G

text: m

text: m

text: Vˆ

text: (s)

text: χˆ

text: χˆ

text: ζˆ

text: t

text: m

text: Provided

text: that

text: µ

text: q,

text: there

text: is

text: a

text: growing-mode

text: solution

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