text: Growth of spherical overdensities in ST gravities 3 Here we will assume the Ratra-Peebles form for the scalar ﬁeld Table 1. Different minimally and non-minimally coupled quintessence potential: models considered in this work. The parameter α indicates the exponent of the inverse power-law potential and ξ represents the strength of the cou- pling between the scalar ﬁeld and gravity. M4+α U(Φ) = , (7) Φα Model ξ α where M is an energy scale and the exponent α is a free pos- Model (1) 0.123 0.261 itive constant. Constraints from magnitude-redshift measurements Model (2) 0.088 0.679 − of type Ia Supernovae show that α < 1 at the 1 σ level both Model (3) 0.000 0.877 for minimally and non-minimally coupled quintessence models (?). Model (4) -0.087 0.877 We further assume a power-law for the the dimensionless function ΛCDM 0.000 0.000 F(Φ): − F(Φ) = 1 + ξ(Φ2 Φ2) , (8) to 1/F(Φ). In middle panel of ﬁgure (1) we show the evolution 0 of the Ratra-Peebles potential for the different models of Table (1). In all the models, the potential decays from larger values at higher where the constant ξ indicates the strength of the coupling be- redshift to one at the present time. In fact we ﬁx the potential to be tween the scalar ﬁeld and Ricci scalar and Φ0 is the value of the one at the present time by setting Φ = 1 and M = 1. We see that scalar ﬁeld at the present time (see also ??). Solar system tests put 0 ≈ − for the non-minimally coupled cases, independently of the sign of very tight constraints on the coupling parameter ξ 10 2, (see ξ, the potential is weaker compared to the minimally coupled case ?????). in GR. A larger value of the coupling parameter ξ causes stronger From equations (3) and (4), the energy density and pressure deviations from the potential in GR gravity. Finally, in the bottom of the scalar ﬁeld in the framework of ST gravities (in the case of panel, we present the redshift evolution of the scalar ﬁeld Φ for dif- Z(Φ) = 1 assumed here) can be easily obtained as follows: ferent models. We see that for all models the scalar ﬁeld increases − with redshift from its small initial value at a = 10 4 to Φ = 1 at i 0 1 − the present time. ρ = Φ˙ 2 + U(Φ) 3HF˙ (Φ) , (9) φ 2 We now calculate the redshift evolution of the background 1 − cosmological parameters: the equation of state of the scalar ﬁeld, p = Φ˙ 2 U(Φ) + F¨(Φ) + 2HF˙ (Φ) . (10) φ 2 w = p /ρ , the energy density parameter of the scalar ﬁeld, Ω Φ Φ Φ Φ and the Hubble expansion rate H, for the different cosmological We now solve numerically the coupled system of equations models considered in this work. As well known, these quantities (3, 4, 5 and 6) for the above potentials U(Φ) and F(Φ). We ﬁx the can describe the background evolution of the universe. Moreover, ≈ − initial time at matter-radiation equality epoch a 10 4 and also the linear growth rate of structures strongly depends on the back- i set the present time values of the scalar ﬁeld and energy density of ground evolution. In ﬁgure (2), using equations (9) and (10), we pressureless dust matter as Φ = 1 and Ω = 0.30, respectively. ﬁrst show the evolution of w as a function of the redshift z (top 0 m,0 Φ We also need two initial conditions to solve the Klein-Gordon equa- panel). We then show the redshift evolution of the energy density tion. Following ?, we set the initial conditions as Φ(a ) = 0.12 Ω for our selected models and Ω for concordance ΛCDM model − i Φ Λ and Φ˙ (a ) = 10 5. We have two free parameters, ξ and α, in this (middle panel). We ﬁnally present the evolution of the fractional i analysis. It should be emphasized that the initial values of these difference of the Hubble parameter ∆H(z) for these models rel- parameters must be chosen such that the function F(Φ) and the atively to the concordance ΛCDM model H (z) in the bot- ΛCDM scalar ﬁeld Φ evolve from their initial values to reach F(Φ) = 1 tom panel of ﬁgure (2). We refer to the caption for the different and Φ = Φ = 1 at the present time. Consequently, the amount colours and line styles. In the case of a positive coupling (ξ > 0), 0 − of energy density of the scalar ﬁeld Ω should be close to 0.70 the phantom regime (w < 1) can be achieved at high redshifts, Φ at the present time. On the basis of different values of the expo- while models with ξ < 0 and ξ = 0 remain in the quintessence − nent parameter of the scalar ﬁeld potential, α, and the strength of regime (w > 1). The amount of DE for different minimally and the non-minimally coupling, ξ, we adopt four different cosmologi- non-minimally coupled quintessence models as well as the ΛCDM cal models in the framework of ST gravities presented in Table (1). model have been presented in the middle panel. For all the models The concordance ΛCDM model is reproduced by setting α = 0 and the amount of DE is negligible at high redshifts meaning that all ξ = 0. Also model (3) represents a minimally coupled quintessence the models reduce to an EdS Universe at early times. The evolution model in which there is no direct coupling between the scalar ﬁeld of ∆H(z) for different models is shown in the bottom panel. As and gravity (ξ = 0). it appears clear, at low redshifts, all the models give ∆H(z) > 0 In ﬁgure (1), we show the evolution of 1/F(Φ) (top panel), indicating that all of them have larger Hubble parameter than the U(Φ) (middle panel) and Φ (bottom panel), as a function of redshift ΛCDM universe. We see that at high redshifts the Hubble param- z for the different cosmological models presented in Table (1). Line eter becomes smaller than the reference one only for a negative styles and colours for each model have been indicated in the cap- coupling constant (model 4). In the case of the minimally coupled tion. While 1/F(Φ) is almost constant at high redshifts and its vari- quintessence (model 3), the Hubble parameter tends to the ﬁdu- ation with cosmic z is very small, it changes rapidly at low redshifts cial value in the ΛCDM model at high redshifts and differences and reaches one at present time. The constant line 1/F(Φ) = 1 become negligible. This result is interesting, because in this model represents model (3), the minimally coupled quintessence model there is no direct coupling between the scalar ﬁeld and gravity and with ξ = 0, as expected. In the next section we will see that the therefore DE models mimics the cosmological constant at high red- effective gravitational constant in ST cosmologies is proportional shifts. In the case of non-minimally coupled quintessence models MNRAS 000, 1–?? (2015) ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Growth

text: of

text: spherical

text: overdensities

text: in

text: ST

text: gravities

text: Here

text: we

text: will

text: assume

text: the

text: Ratra-Peebles

text: form

text: for

text: the

text: scalar

text: ﬁeld

text: Table

text: Different

text: minimally

text: and

text: non-minimally

text: coupled

text: quintessence

text: potential:

text: models

text: considered

text: in

text: this

text: work.

text: The

text: parameter

text: α

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text: exponent

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text: inverse

text: power-law

text: potential

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text: ξ

text: represents

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text: strength

text: of

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text: cou-

text: pling

text: between

text: the

text: scalar

text: ﬁeld

text: and

text: gravity.

label: equation

text: U(Φ)

text: Φα

text: Model

label: table

text: ξ

text: α

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text: M

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text: exponent

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text: Model

text: itive

text: constant.

text: Constraints

text: from

text: magnitude-redshift

text: measurements

text: Model

text: of

text: type

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text: show

text: that

text: α

text: at

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text: σ

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text: ﬁgure

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text: equations

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text: ΛCDM

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text: panel

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text: refer

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text: the

text: caption

text: for

text: the

text: different

text: and

text: Φ

text: Φ

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text: the

text: present

text: time.

text: Consequently,

text: the

text: amount

text: colours

text: and

text: line

text: styles.

text: In

text: the

text: case

text: of

text: a

text: positive

text: coupling

text: (ξ

text: of

text: energy

text: density

text: of

text: the

text: scalar

text: ﬁeld

text: Ω

text: should

text: be

text: close

text: to

text: the

text: phantom

text: regime

text: (w

text: can

text: be

text: achieved

text: at

text: high

text: redshifts,

text: Φ

text: at

text: the

text: present

text: time.

text: On

text: the

text: basis

text: of

text: different

text: values

text: of

text: the

text: expo-

text: while

text: models

text: with

text: ξ

text: and

text: ξ

text: remain

text: in

text: the

text: quintessence

text: nent

text: parameter

text: of

text: the

text: scalar

text: ﬁeld

text: potential,

text: α,

text: and

text: the

text: strength

text: of

text: regime

text: (w

text: The

text: amount

text: of

text: DE

text: for

text: different

text: minimally

text: and

text: the

text: non-minimally

text: coupling,

text: ξ,

text: we

text: adopt

text: four

text: different

text: cosmologi-

text: non-minimally

text: coupled

text: quintessence

text: models

text: as

text: well

text: as

text: the

text: ΛCDM

text: cal

text: models

text: in

text: the

text: framework

text: of

text: ST

text: gravities

text: presented

text: in

text: Table

text: model

text: have

text: been

text: presented

text: in

text: the

text: middle

text: panel.

text: For

text: all

text: the

text: models

text: The

text: concordance

text: ΛCDM

text: model

text: is

text: reproduced

text: by

text: setting

text: α

text: and

text: the

text: amount

text: of

text: DE

text: is

text: negligible

text: at

text: high

text: redshifts

text: meaning

text: that

text: all

text: ξ

text: Also

text: model

text: represents

text: a

text: minimally

text: coupled

text: quintessence

text: the

text: models

text: reduce

text: to

text: an

text: EdS

text: Universe

text: at

text: early

text: times.

text: The

text: evolution

text: model

text: in

text: which

text: there

text: is

text: no

text: direct

text: coupling

text: between

text: the

text: scalar

text: ﬁeld

text: of

text: ∆H(z)

text: for

text: different

text: models

text: is

text: shown

text: in

text: the

text: bottom

text: panel.

text: As

text: and

text: gravity

text: (ξ

text: it

text: appears

text: clear,

text: at

text: low

text: redshifts,

text: all

text: the

text: models

text: give

text: ∆H(z)

text: In

text: ﬁgure

text: we

text: show

text: the

text: evolution

text: of

text: (top

text: panel),

text: indicating

text: that

text: all

text: of

text: them

text: have

text: larger

text: Hubble

text: parameter

text: than

text: the

text: U(Φ)

text: (middle

text: panel)

text: and

text: Φ

text: (bottom

text: panel),

text: as

text: a

text: function

text: of

text: redshift

text: ΛCDM

text: universe.

text: We

text: see

text: that

text: at

text: high

text: redshifts

text: the

text: Hubble

text: param-

text: z

text: for

text: the

text: different

text: cosmological

text: models

text: presented

text: in

text: Table

text: Line

text: eter

text: becomes

text: smaller

text: than

text: the

text: reference

text: one

text: only

text: for

text: a

text: negative

text: styles

text: and

text: colours

text: for

text: each

text: model

text: have

text: been

text: indicated

text: in

text: the

text: cap-

text: coupling

text: constant

text: (model

text: In

text: the

text: case

text: of

text: the

text: minimally

text: coupled

text: tion.

text: While

text: is

text: almost

text: constant

text: at

text: high

text: redshifts

text: and

text: its

text: vari-

text: quintessence

text: (model

text: the

text: Hubble

text: parameter

text: tends

text: to

text: the

text: ﬁdu-

text: ation

text: with

text: cosmic

text: z

text: is

text: very

text: small,

text: it

text: changes

text: rapidly

text: at

text: low

text: redshifts

text: cial

text: value

text: in

text: the

text: ΛCDM

text: model

text: at

text: high

text: redshifts

text: and

text: differences

text: and

text: reaches

text: one

text: at

text: present

text: time.

text: The

text: constant

text: line

text: become

text: negligible.

text: This

text: result

text: is

text: interesting,

text: because

text: in

text: this

text: model

text: represents

text: model

text: the

text: minimally

text: coupled

text: quintessence

text: model

text: there

text: is

text: no

text: direct

text: coupling

text: between

text: the

text: scalar

text: ﬁeld

text: and

text: gravity

text: and

text: with

text: ξ

text: as

text: expected.

text: In

text: the

text: next

text: section

text: we

text: will

text: see

text: that

text: the

text: therefore

text: DE

text: models

text: mimics

text: the

text: cosmological

text: constant

text: at

text: high

text: red-

text: effective

text: gravitational

text: constant

text: in

text: ST

text: cosmologies

text: is

text: proportional

text: shifts.

text: In

text: the

text: case

text: of

text: non-minimally

text: coupled

text: quintessence

text: models

text: MNRAS

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 10 Nazari-Pooya et al. Table 2. Numerical results for the fractional difference of number of haloes Table 3. The fractional difference of number of haloes between clustering between homogeneous minimally and non-minimally coupled quintessence models in ST cosmologies and the concordance ΛCDM model. Results are models and the concordance ΛCDM model. These results are presented at shown at four different epochs: z = 0, z = 0.5, z = 1 and z = 2 for ob- four different redshifts: z = 0, z = 0.5, z = 1 and z = 2 for objects jects with M > 1013M(cid:12)/h, M > 1014M(cid:12)/h and M > 1015M(cid:12)/h. with M > 1013M(cid:12)/h (low mass scale), M > 1014M(cid:12)/h (intermediate mass scale) and M > 1015M(cid:12)/h (high mass end). Model 1(b) z = 0 z = 0.5 z = 1 z = 2 Model 1(a) z = 0 z = 0.5 z = 1 z = 2 M > 1013M(cid:12)/h -2.6% -3.2% -4.5% -6.4% M > 1014M(cid:12)/h -2.7% -3.6% -5.3% -12% M > 1013M(cid:12)/h -0.1% 0.3% 0.9% 3% M > 1015M(cid:12)/h -3.1% -5% -11.8% -34% M > 1014M(cid:12)/h -0.2% 0.6% 2% 6% M > 1015M(cid:12)/h -0.6% 2% 7% 21% Model 2(b) z = 0 z = 0.5 z = 1 z = 2 Model 2(a) z = 0 z = 0.5 z = 1 z = 2 M > 1013M(cid:12)/h -1.4% 0.6% 3.8% 19% M > 1014M(cid:12)/h -1.5% 1.9% 8.5% 39% M > 1013M(cid:12)/h -0.1% 1.2% 5% 22% M > 1015M(cid:12)/h -1.9% 7.7% 32% - M > 1014M(cid:12)/h -0.2% 3% 11% 47% M > 1015M(cid:12)/h -0.6% 11% 40% - Model 3 z = 0 z = 0.5 z = 1 z = 2 Model 3 z = 0 z = 0.5 z = 1 z = 2 M > 1013M(cid:12)/h -0.1% 2% 8% 39% M > 1014M(cid:12)/h -0.2% 4% 17% 89% M > 1013M(cid:12)/h -0.1% 2% 8% 39% M > 1015M(cid:12)/h -0.5% 17% 71% - M > 1014M(cid:12)/h -0.2% 4% 17% 89% M > 1015M(cid:12)/h -0.5% 17% 71% - Model 4(b) z = 0 z = 0.5 z = 1 z = 2 Model 4(a) z = 0 z = 0.5 z = 1 z = 2 M > 1013M(cid:12)/h -0.8% 2% 8.5% 42% M > 1014M(cid:12)/h -1% 5% 19% 95% M > 1013M(cid:12)/h -0.1% 2.3% 9% 43% M > 1015M(cid:12)/h -1.6% 20% 81% - M > 1014M(cid:12)/h -0.2% 6% 21% 101% M > 1015M(cid:12)/h -0.5% 22% 89% - ≈ at z 0, all models roughly coincide with the ΛCDM case and In ﬁgure (??), we show the redshift evolution of (cid:15)(z) on differences are negligible. At z = 0.5 differences with the ΛCDM the basis of equation (29) for the different minimally and non- model are considerable at the high mass tail (M > 1015M(cid:12)/h). minimally coupled quintessence models considered in this work. In particular, we see that for model (4b) the number of haloes is We see that for all the non-minimally coupled models with posi- roughly 20% higher than that of the concordance ΛCDM model. tive or negative coupling constant ξ, the quantity (cid:15) is negative and This valu−e is 17% for model 3 and approximately 8% for model for the minimally coupled case (model 3) (cid:15) is zero. This result is (2b) and 5% in the case of model (1b). The numerical values for expected, since δ for all non-minimally coupled models is nega- the fractional difference of the number of haloes between the dif- tive and zero for Φmodel 3 (see also ﬁgure 4). We also see that for ferent clustering models considered in this work and the ΛCDM z (cid:38) 1, in all non-minimally coupled models (cid:15) approaches zero. model are presented in table (3). In analogy to table (2), the re- This means that the contribution of the DE mass to the total mass sults are reported for masses greater than 1013M(cid:12)/h, 1014M(cid:12)/h of haloes is negligible at high redshifts. This is nothing else than re- and 1015M(cid:12)/h. At higher redshifts, z = 1 and z = 2, we see quiring an EdS behaviour in the past. Moreover, models with larger that the predicted number of haloes calculated for clustering mod- coupling constant ξ give a higher contribution of the DE mass to the els deviates from the ΛCDM expectation even at the low mass end total mass of haloes. We notice that in these models the DE mass of the halo mass function. Generally, by comparing the results pre- is negative and hence should be subtracted from the total mass of sented in tables (2) and (3), we conclude that the number of viri- haloes. alised haloes estimated in clustering models is lower than that for Following ???, in the presence of DE mass contribution, the homogeneous models. This result is due to the negative sign of δΦ deﬁnition of the mass function in the Press-Schechter formalism is and (cid:15) in clustering quintessence models. Differences between clus- changed to tering and homogeneous models are more pronounced at the high mass tail and at high redshifts, as expected. For example for model (1) where the coupling constant is the largest (ξ = 0.123), we see dn(M, w, z) ρ¯ dν(M, w, z) = 0− f(ν) , (30) that at the high mass tail (M > 1015M(cid:12)/h at z = 0) the homo- dM M(1 (cid:15)) dM geneous model (1a) produces only 2.5% more objects than in the clustering case (model (1b)). This value is 7% at z = 0.5, roughly where f(ν) is given by equation (26). We notice that in clus- 19% at z = 1 and 55% at z = 2. For other quintessence models tering models, density perturbations of the scalar ﬁeld change the we have similar results. function f(ν) via changing the quantities δc and σ. In ﬁgure (8), 5 CONCLUSIONS we show the predicted number of haloes calculated with the cor- rected mass function in equation (30) for clustering models divided In the context of the spherical collapse model (SCM) we stud- by those obtained for the concordance ΛCDM model. In analogy ied the non-linear growth of structures in scalar-tensor (ST) cos- to ﬁgure (7), we select four different redshifts z = 0, z = 0.5, mologies. In ST gravity, there is a non-minimally coupling between z = 1 and z = 2. As in the previous section, clustering models the scalar ﬁeld and the Ricci scalar, the so-called non-minimally are labelled with the letter "(b)". Results for model (3) are similar coupled quintessence models. We ﬁrst studied the background ex- to those obtained before and presented in section (). We see that pansion history in ST theories. We saw that in the case of positive MNRAS 000, 1–?? (2015) ##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##

label: paragraph

text: Nazari-Pooya

text: et

text: al.

text: Table

text: Numerical

label: caption

text: results

text: for

text: the

text: fractional

text: difference

text: of

text: number

text: of

text: haloes

text: Table

text: The

text: fractional

text: difference

text: of

text: number

text: of

text: haloes

text: between

text: clustering

text: between

text: homogeneous

text: minimally

text: and

text: non-minimally

text: coupled

text: quintessence

text: models

text: in

text: ST

text: cosmologies

text: and

text: the

text: concordance

text: ΛCDM

text: model.

text: Results

text: are

text: models

text: and

text: the

text: concordance

text: ΛCDM

text: model.

text: These

text: results

text: are

text: presented

text: at

text: shown

text: at

text: four

text: different

text: epochs:

text: z

text: z

text: z

text: and

text: z

text: for

text: ob-

text: four

text: different

text: redshifts:

text: z

text: z

text: z

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text: z

text: for

text: objects

text: jects

text: with

text: M

text: M

text: and

text: M

text: with

text: M

text: (low

text: mass

text: scale),

text: M

text: (intermediate

text: mass

text: scale)

text: and

text: M

text: (high

text: mass

text: end).

text: Model

text: z

text: z

text: z

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text: z

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text: M

text: M

text: M

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text: M

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text: z

text: z

text: z

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text: z

text: z

text: z

text: z

text: M

text: M

text: M

text: M

text: M

text: M

text: at

text: z

text: all

text: models

text: roughly

text: coincide

text: with

text: the

text: ΛCDM

text: case

text: and

text: In

text: ﬁgure

text: we

text: show

text: the

text: redshift

text: evolution

text: of

text: on

text: differences

text: are

text: negligible.

text: At

text: z

text: differences

text: with

text: the

text: ΛCDM

text: the

text: basis

text: of

text: equation

text: for

text: the

text: different

text: minimally

text: and

text: non-

text: model

text: are

text: considerable

text: at

text: the

text: high

text: mass

text: tail

text: (M

text: minimally

text: coupled

text: quintessence

text: models

text: considered

text: in

text: this

text: work.

text: In

text: particular,

text: we

text: see

text: that

text: for

text: model

text: the

text: number

text: of

text: haloes

text: is

text: We

text: see

text: that

text: for

text: all

text: the

text: non-minimally

text: coupled

text: models

text: with

text: posi-

text: roughly

text: higher

text: than

text: that

text: of

text: the

text: concordance

text: ΛCDM

text: model.

text: tive

text: or

text: negative

text: coupling

text: constant

text: ξ,

text: the

text: quantity

text: is

text: negative

text: and

text: This

text: valu−e

text: is

text: for

text: model

text: and

text: approximately

text: for

text: model

text: for

text: the

text: minimally

text: coupled

text: case

text: (model

text: is

text: zero.

text: This

text: result

text: is

text: and

text: in

text: the

text: case

text: of

text: model

text: The

text: numerical

text: values

text: for

text: expected,

text: since

text: δ

text: for

text: all

text: non-minimally

text: coupled

text: models

text: is

text: nega-

text: the

text: fractional

text: difference

text: of

text: the

text: number

text: of

text: haloes

text: between

text: the

text: dif-

text: tive

text: and

text: zero

text: for

text: Φmodel

text: (see

text: also

text: ﬁgure

text: We

text: also

text: see

text: that

text: for

text: ferent

text: clustering

text: models

text: considered

text: in

text: this

text: work

text: and

text: the

text: ΛCDM

text: z

text: in

text: all

text: non-minimally

text: coupled

text: models

text: approaches

text: zero.

text: model

text: are

text: presented

text: in

text: table

text: In

text: analogy

text: to

text: table

text: the

text: re-

text: This

text: means

text: that

text: the

text: contribution

text: of

text: the

text: DE

text: mass

text: to

text: the

text: total

text: mass

text: sults

text: are

text: reported

text: for

text: masses

text: greater

text: than

text: of

text: haloes

text: is

text: negligible

text: at

text: high

text: redshifts.

text: This

text: is

text: nothing

text: else

text: than

text: re-

text: and

text: At

text: higher

text: redshifts,

text: z

text: and

text: z

text: we

text: see

text: quiring

text: an

text: EdS

text: behaviour

text: in

text: the

text: past.

text: Moreover,

text: models

text: with

text: larger

text: that

text: the

text: predicted

text: number

text: of

text: haloes

text: calculated

text: for

text: clustering

text: mod-

text: coupling

text: constant

text: ξ

text: give

text: a

text: higher

text: contribution

text: of

text: the

text: DE

text: mass

text: to

text: the

text: els

text: deviates

text: from

text: the

text: ΛCDM

text: expectation

text: even

text: at

text: the

text: low

text: mass

text: end

text: total

text: mass

text: of

text: haloes.

text: We

text: notice

text: that

text: in

text: these

text: models

text: the

text: DE

text: mass

text: of

text: the

text: halo

text: mass

text: function.

text: Generally,

text: by

text: comparing

text: the

text: results

text: pre-

text: is

text: negative

text: and

text: hence

text: should

text: be

text: subtracted

text: from

text: the

text: total

text: mass

text: of

text: sented

text: in

text: tables

text: and

text: we

text: conclude

text: that

text: the

text: number

text: of

text: viri-

text: haloes.

text: alised

text: haloes

text: estimated

text: in

text: clustering

text: models

text: is

text: lower

text: than

text: that

text: for

text: Following

text: in

text: the

text: presence

text: of

text: DE

text: mass

text: contribution,

text: the

text: homogeneous

text: models.

text: This

text: result

text: is

text: due

text: to

text: the

text: negative

text: sign

text: of

text: δΦ

text: deﬁnition

text: of

text: the

text: mass

text: function

text: in

text: the

text: Press-Schechter

text: formalism

text: is

text: and

text: in

text: clustering

text: quintessence

text: models.

text: Differences

text: between

text: clus-

text: changed

text: to

text: tering

text: and

text: homogeneous

text: models

text: are

text: more

text: pronounced

text: at

text: the

text: high

text: mass

text: tail

text: and

text: at

text: high

text: redshifts,

text: as

text: expected.

text: For

text: example

text: for

text: model

text: where

text: the

text: coupling

text: constant

text: is

text: the

text: largest

text: (ξ

text: we

text: see

text: dn(M,

text: w,

text: z)

text: ρ¯

text: dν(M,

text: w,

text: z)

text: 0−

text: f(ν)

text: that

text: at

text: the

text: high

text: mass

text: tail

text: (M

text: at

text: z

text: the

text: homo-

text: dM

text: dM

text: geneous

text: model

text: produces

text: only

text: more

text: objects

text: than

text: in

text: the

text: clustering

text: case

text: (model

text: This

text: value

text: is

text: at

text: z

text: roughly

text: where

text: f(ν)

text: is

text: given

text: by

text: equation

text: We

text: notice

text: that

text: in

text: clus-

text: at

text: z

text: and

text: at

text: z

text: For

text: other

text: quintessence

text: models

text: tering

text: models,

text: density

text: perturbations

text: of

text: the

text: scalar

text: ﬁeld

text: change

text: the

text: we

text: have

text: similar

text: results.

text: function

text: f(ν)

text: via

text: changing

text: the

text: quantities

text: δc

text: and

text: σ.

text: In

text: ﬁgure

text: CONCLUSIONS

text: we

text: show

text: the

text: predicted

text: number

text: of

text: haloes

text: calculated

text: with

text: the

text: cor-

text: rected

text: mass

text: function

text: in

text: equation

text: for

text: clustering

text: models

text: divided

text: In

text: the

text: context

text: of

text: the

text: spherical

text: collapse

text: model

text: (SCM)

text: we

text: stud-

text: by

text: those

text: obtained

text: for

text: the

text: concordance

text: ΛCDM

text: model.

text: In

text: analogy

text: ied

text: the

text: non-linear

text: growth

text: of

text: structures

text: in

text: scalar-tensor

text: (ST)

text: cos-

text: to

text: ﬁgure

text: we

text: select

text: four

text: different

text: redshifts

text: z

text: z

text: mologies.

text: In

text: ST

text: gravity,

text: there

text: is

text: a

text: non-minimally

text: coupling

text: between

text: z

text: and

text: z

text: As

text: in

text: the

text: previous

text: section,

text: clustering

text: models

text: the

text: scalar

text: ﬁeld

text: and

text: the

text: Ricci

text: scalar,

text: the

text: so-called

text: non-minimally

text: are

text: labelled

text: with

text: the

text: letter

text: "(b)".

text: Results

text: for

text: model

text: are

text: similar

text: coupled

text: quintessence

text: models.

text: We

text: ﬁrst

text: studied

text: the

text: background

text: ex-

text: to

text: those

text: obtained

text: before

text: and

text: presented

text: in

text: section

text: We

text: see

text: that

text: pansion

text: history

text: in

text: ST

text: theories.

text: We

text: saw

text: that

text: in

text: the

text: case

text: of

text: positive

text: MNRAS

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text: 2 1 − − by dividing α¯i with kdi. The ﬂow equations of the dimen- Γ˙ [g¯, φ] = sionful couplings α¯ (k) of mass dimension d are related k i i 2 to the ﬂow of their dimensionless counterparts α (k) by i − FIG. 1. Flow equation for the scale dependent eﬀective ac- k diα¯˙ (k) = d α + α˙ . (4) i i i i tion Γ in diagrammatic representation. The double, dotted, k and solid lines correspond to the graviton, ghost, and fermion propagators, respectively. The crossed circles denote the re- The truncation applied in this work is constructed spective regulator insertions. along the same lines as in [1, 2, 8]. Thus, Γ [g¯, φ] is k given in terms of a vertex expansion in the ﬂuctuat- ing ﬁelds φ about the expansion point φ = 0 with the ﬁeld g is split linearly into background and ﬂuctuating µν structure (cid:88) ﬁelds, g¯ and h , respectively. For the present theory µν µν of gravity and fermions, the complete set of ﬂuctuating ∞ ﬁelds is given by 1 Γ [g¯, φ] = Γ(n)[g¯, 0] ◦ φn , (5) k n! k n=0 φ = (h, c¯, c, ψ¯, ψ) . (1) where the circle indicates the integration of φ(x ) · · · φ(x ) 1 n with Γ(n)(x , ..., x ). As can be seen from (2), the two- 1 n Here, (c¯, c) denote the (anti-)ghosts and (ψ¯, ψ) are the point functions plays a pivotal rˆole in the current ap- proach. We parameterise (anti-)fermion ﬁelds. By adding a scale dependent IR- regulator term R [g¯, φ] to the classical action we obtain k the scale-dependent eﬀective action Γk[g¯, φ]. It gives Γ(φφ)(p2) = Z (p2)T (2) , (6) an eﬀective description of the physics at scale k in the k φ spirit of the Wilsonian renormalisation group. The ﬂow of Γk[g¯, φ] is governed by the Wetterich equation [42], where T (2) carries the tensor structure of the two-point applied to gravity [4]. For the given ﬁeld content (1) it function, see (8), and Z are the momentum-dependent φ reads (cid:34) (cid:35) wave function renormalisations. Similarly, the vertices Γ(n) are parame(cid:32)terised as (cid:33) (cid:32) (cid:33) k (cid:89) (cid:113) (cid:89) (cid:113) 1 1 Γ˙ [g¯, φ] =(cid:34) Tr (cid:35) R˙ (cid:34) (cid:35) k 2 Γ(k2) + Rk k hh Γ(n)[g¯, 0] = nh Z (p2 ) nc Z (p2 ) (2) k (cid:32) h h,i (cid:33) c c,i i=1(cid:89) (cid:113) i=1 1 1 −Tr R˙ − Tr R˙ , (2) k (2) k Γk + Rk c¯c Γk + Rk ψ¯ψ nψ − × Z (p2 ) G12(n 2)T (n) , (7) ψ ψ,i (cid:126)n where Tr denotes the summation over discrete and in- i=1 tegration of continuous variables. We abbreviate by a where (cid:126)n = (n , n , n ) and n = n + n + n . Here, dot derivatives with respect to t = log(k/k0), where k0 is h c ψ h c ψ n , n and n denote the number of graviton, ghost some arbitrary reference scale. Figure 1 depicts equation h c ψ and fermion legs, respectively. The wave function renor- (2) in terms of diagrams. From now on, we will drop the malisations in (7) carry the RG-running of the vertices. regulator insertions in the diagrams as well as the arrows Indeed, for the two-point functions (7) boils down to for Grassmann-valued ﬁelds for convenience. (6). For n (cid:54)= 2 the G are renormalisation group in- Eq. (2) cannot be solved for Γ [g¯, φ] in full generality. (cid:126)n k variant couplings. For example, for the pure gravita- Therefore, the eﬀective action is typically truncated to a tional vertices, n = n + n the G are momentum de- ﬁnite set of functionals O to wit h c (cid:126)n i (cid:88) pendent Newton’s constants G (p). Here, p is the vec- n tor of all ﬁeld momenta. The vertices that implement N the minimal coupling between gravity and fermions are Γ [g¯, φ] = α¯ (k)O [g¯, φ] . (3) consequently associated with G (p). In the same k i i nh,0,2 i=0 way as in [1, 2], however, we approximate here all Gn as one, momentum-independent coupling, G (p) = nh,nc,0 With help of this ansatz the ﬂow equation (2) provides Gnh,0,2(p) ≡ G3,0,0 =: G. In the four-fermion case we a ﬁnite dimensional coupled system of ﬂow equations for have Gn,0,4 = λψ. the k-dependent dimensionful couplings α¯ (k) with mass The tensor structures T (n) are given by the nth varia- i dimension d . Dimensionless couplings α are introduced tion of the classical action S to be speciﬁed below. The i i ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: by

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text: with

text: kdi.

text: The

text: ﬂow

text: equations

text: of

text: the

text: dimen-

text: Γ˙

text: [g¯,

text: φ]

text: sionful

text: couplings

text: α¯

text: (k)

text: of

text: mass

text: dimension

text: d

text: are

text: related

text: k

text: i

text: i

text: to

text: the

text: ﬂow

text: of

text: their

text: dimensionless

text: counterparts

text: α

text: (k)

text: by

text: i

text: FIG.

text: Flow

text: equation

text: for

text: the

text: scale

text: dependent

text: eﬀective

text: ac-

text: k

text: diα¯˙

text: (k)

text: d

text: α

text: α˙

text: i

text: i

text: i

text: i

text: tion

text: Γ

text: in

text: diagrammatic

text: representation.

text: The

text: double,

text: dotted,

text: k

text: and

text: solid

text: lines

text: correspond

text: to

text: the

text: graviton,

text: ghost,

text: and

text: fermion

text: propagators,

text: respectively.

text: The

text: crossed

text: circles

text: denote

text: the

text: re-

text: The

text: truncation

text: applied

text: in

text: this

text: work

text: is

text: constructed

text: spective

text: regulator

text: insertions.

text: along

text: the

text: same

text: lines

text: as

text: in

text: Thus,

text: Γ

text: [g¯,

text: φ]

text: is

text: k

text: given

text: in

text: terms

text: of

text: a

text: vertex

text: expansion

text: in

text: the

text: ﬂuctuat-

text: ing

text: ﬁelds

text: φ

text: about

text: the

text: expansion

text: point

text: φ

text: with

text: the

text: ﬁeld

text: g

text: is

text: split

text: linearly

text: into

text: background

text: and

text: ﬂuctuating

text: µν

text: structure

text: ﬁelds,

text: g¯

text: and

text: h

text: respectively.

text: For

text: the

text: present

text: theory

text: µν

text: µν

text: of

text: gravity

text: and

text: fermions,

text: the

text: complete

text: set

text: of

text: ﬂuctuating

text: ﬁelds

text: is

text: given

text: by

text: Γ

text: [g¯,

text: φ]

text: Γ(n)[g¯,

text: φn

text: k

text: n!

text: k

text: φ

text: (h,

text: c¯,

text: c,

text: ψ¯,

text: ψ)

text: where

text: the

text: circle

text: indicates

text: the

text: integration

text: of

text: φ(x

text: φ(x

text: n

text: with

text: Γ(n)(x

text: x

text: As

text: can

text: be

text: seen

text: from

text: the

text: two-

text: n

text: Here,

text: (c¯,

text: c)

text: denote

text: the

text: (anti-)ghosts

text: and

text: (ψ¯,

text: ψ)

text: are

text: the

text: point

text: functions

text: plays

text: a

text: pivotal

text: rˆole

text: in

text: the

text: current

text: ap-

text: proach.

text: We

text: parameterise

text: (anti-)fermion

text: ﬁelds.

text: By

text: adding

text: a

text: scale

text: dependent

text: IR-

text: regulator

text: term

text: R

text: [g¯,

text: φ]

text: to

text: the

text: classical

text: action

text: we

text: obtain

text: k

text: the

text: scale-dependent

text: eﬀective

text: action

text: Γk[g¯,

text: φ].

text: It

text: gives

text: Z

text: an

text: eﬀective

text: description

text: of

text: the

text: physics

text: at

text: scale

text: k

text: in

text: the

text: k

text: φ

text: spirit

text: of

text: the

text: Wilsonian

text: renormalisation

text: group.

text: The

text: ﬂow

text: of

text: Γk[g¯,

text: φ]

text: is

text: governed

text: by

text: the

text: Wetterich

text: equation

text: where

text: T

text: carries

text: the

text: tensor

text: structure

text: of

text: the

text: two-point

text: applied

text: to

text: gravity

text: For

text: the

text: given

text: ﬁeld

text: content

text: it

text: function,

text: see

text: and

text: Z

text: are

text: the

text: momentum-dependent

text: φ

text: reads

text: wave

text: function

text: renormalisations.

text: Similarly,

text: the

text: vertices

text: Γ(n)

text: are

text: as

text: k

text: Γ˙

text: [g¯,

text: φ]

text: Tr

text: R˙

text: k

text: Rk

text: k

text: hh

text: Γ(n)[g¯,

text: nh

text: Z

text: nc

text: Z

text: k

text: h

text: h,i

text: c

text: c,i

text: −Tr

text: R˙

text: Tr

text: R˙

text: k

text: k

text: Γk

text: Rk

text: c¯c

text: Γk

text: Rk

text: ψ¯ψ

text: nψ

text: Z

text: (n)

text: ψ

text: ψ,i

text: where

text: Tr

text: denotes

text: the

text: summation

text: over

text: discrete

text: and

text: in-

text: tegration

text: of

text: continuous

text: variables.

text: We

text: abbreviate

text: by

text: a

text: where

text: (n

text: n

text: n

text: and

text: n

text: n

text: n

text: n

text: Here,

text: dot

text: derivatives

text: with

text: respect

text: to

text: t

text: where

text: is

text: h

text: c

text: ψ

text: h

text: c

text: ψ

text: n

text: n

text: and

text: n

text: denote

text: the

text: number

text: of

text: graviton,

text: ghost

text: some

text: arbitrary

text: reference

text: scale.

text: Figure

text: depicts

text: equation

text: h

text: c

text: ψ

text: and

text: fermion

text: legs,

text: respectively.

text: The

text: wave

text: function

text: renor-

text: in

text: terms

text: of

text: diagrams.

text: From

text: now

text: on,

text: we

text: will

text: drop

text: the

text: malisations

text: in

text: carry

text: the

text: RG-running

text: of

text: the

text: vertices.

text: regulator

text: insertions

text: in

text: the

text: diagrams

text: as

text: well

text: as

text: the

text: arrows

text: Indeed,

text: for

text: the

text: two-point

text: functions

text: boils

text: down

text: to

text: for

text: Grassmann-valued

text: ﬁelds

text: for

text: convenience.

text: For

text: n

text: the

text: G

text: are

text: renormalisation

text: group

text: in-

text: Eq.

text: cannot

text: be

text: solved

text: for

text: Γ

text: [g¯,

text: φ]

text: in

text: full

text: generality.

text: k

text: variant

text: couplings.

text: For

text: example,

text: for

text: the

text: pure

text: gravita-

text: Therefore,

text: the

text: eﬀective

text: action

text: is

text: typically

text: truncated

text: to

text: a

text: tional

text: vertices,

text: n

text: n

text: n

text: the

text: G

text: are

text: momentum

text: de-

text: ﬁnite

text: set

text: of

text: functionals

text: O

text: to

text: wit

text: h

text: c

text: i

text: pendent

text: Newton’s

text: constants

text: G

text: (p).

text: Here,

text: p

text: is

text: the

text: vec-

text: n

text: tor

text: of

text: all

text: ﬁeld

text: momenta.

text: The

text: vertices

text: that

text: implement

text: N

text: the

text: minimal

text: coupling

text: between

text: gravity

text: and

text: fermions

text: are

text: Γ

text: [g¯,

text: φ]

text: α¯

text: (k)O

text: [g¯,

text: φ]

text: consequently

text: associated

text: with

text: G

text: (p).

text: In

text: the

text: same

text: k

text: i

text: i

text: way

text: as

text: in

text: however,

text: we

text: approximate

text: here

text: all

text: Gn

text: as

text: one,

text: momentum-independent

text: coupling,

text: G

text: (p)

text: With

text: help

text: of

text: this

text: ansatz

text: the

text: ﬂow

text: equation

text: provides

text: G.

text: In

text: the

text: four-fermion

text: case

text: we

text: a

text: ﬁnite

text: dimensional

text: coupled

text: system

text: of

text: ﬂow

text: equations

text: for

text: have

text: λψ.

text: the

text: k-dependent

text: dimensionful

text: couplings

text: α¯

text: (k)

text: with

text: mass

text: The

text: tensor

text: structures

text: T

text: (n)

text: are

text: given

text: by

text: the

text: nth

text: varia-

text: i

text: dimension

text: d

text: Dimensionless

text: couplings

text: α

text: are

text: introduced

text: tion

text: of

text: the

text: classical

text: action

text: S

text: to

text: be

text: speciﬁed

text: below.

text: The

text: i

text: i

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 In particular, the fermion anomalous dimension at van- The sign of F in the deep UV is not ﬁxed for general ishing momentum η (0) as well as f do not play any rˆole regulators. However, we will show that deﬁnite state- ψ for the sign of left-hand side of (22) since they enter ments about the latter are indeed possible once a regula- √ quadratically. We have shown that the annihilation of tor is chosen. In the following, we consider x rψ(x) = √ k ﬁxed points due to gravitational interactions in our model (1 − x)θ(1 − x) and x rh(x) = (1 − x)θ(1 − x) which k is impossible. The only possible way to destroy the well- allow for analytic ﬂow equations. For these choices, F is deﬁned limits now, is via the change of stability of the given by (cid:18) (cid:19) ﬁxed points. B. Stability of Fixed Points g 448 − 79η (k2) 7 − 2η (k2) F = h − ψ + η (0) + 1 , 70π 8(1 + µ)2 1 + µ ψ We now turn to our second criterion, the stability of the ﬁxed points in the NJL-model. It is conceivable, that, (25) if the gravitational interaction is strong enough, it might change the attractivity of its ﬁ(cid:16)xed p(cid:17)oints. In the vicinity where we have moved the anomalous dimensions out of ﬁxed points, the ﬂow can be linearised. Thus, its be- of the loop integrals at momenta at which the integrand haviour is described by critical exponents θ which are is peaked, i.e. q2 = k2. This is shown to be a good n the eigenvalues of the ma(cid:34)tr(cid:18)ix ∂β(cid:19)λi(cid:12) to(cid:35) wit approximation in [2, 8]. Now we investigate, for which ∂λj (cid:12) values of 1 + µ, F does become zero, and, thus where (cid:12) (cid:12) F < 0 is in principle possible. For the following analysis, ∂β we make use of the bounds of the anomalous dimensions θn = Spec ∂λλi F . (23) ηψ(p2) < 1 and ηh(p2) < 2, which are necessary such j n that the corresponding class of regulators, (10), remains well-behaved, for details see [2]. We will take η (p2) as ψ Hence, from the ﬂow equations (19), together with the momentum-independent and write η = η (p2). Further- φ φ (cid:47) ﬁxed point values (21) we ﬁnd the critical exponents of more, we assume that g does not take large values g 10 the NJL-(cid:40)model w(cid:114)ith gravity. (cid:114)They take(cid:41)the simple form along the ﬂow from k = ∞ to k → 0. This assumption is supported by the FRG studies of pure quantum grav- ity and quantum gravity with the given matter content hg hg θ1 = (cid:40)6F − 4(cid:114)F2 + , 2 (cid:114)F2 + (cid:41), Nf = 1 known to the authors. Summarising the main 2 2 steps, we solve F = 0 for 1 + µ, where F given in (25). Employing the approximation for η (p2) = η and the ψ ψ hg hg regulator constraints ﬁrst leads to θ2 = (cid:40)6F + 4 F2 +(cid:112) , −2 F2 + , (cid:41) (24) 2 2 −1 < η < 1 . (26) ψ F2 + 2hg ± 81F4 + 4hg(hg − 7F2) θ = . 3 4F The assumption that g does not become large, implies that η must be very close to the lower limit in (26). We ψ Considering again the limit of vanishing gravitational express this ﬁnding as coupling g → 0, thus F → 1 and g → 0, we get θ = 1 {2, 2}, θ = {10, −2} and θ = {5/2, −2} which we know 2 3 from the NJL-model without gravity. Note that these 1 + ηψ =: δ (cid:28) 1 , (27) quantities are regulator independent in the limit g → 0, since the fermion anomalous dimension does not depend which allows to expand the equation for 1 + µ in a on the 4-fermion couplings. Hence, this fact is a conse- Taylor series in δ. More details can be found in Appendix quence of the momentum independent 4-fermion interac- . The analysis results in a lower bound for 1 + µ, namely tions considered here. For g (cid:54)= 0 however, the regulator dependent quantities F, h and g enter the equations (24). From these equations we can see that na¨ıvely, the θ could 145 i 1 + µ > > 4 . (28) ﬂip sign, if F becomes negative. Since F = 1 + ηψ(0) + f 36 and neither η (p2) nor f have a deﬁnite sign, the case ψ F < 0 is not excluded generally. However, since (η , f) → ψ Since all analyses of the UV behavior of quantum grav- (0, 0) for g → 0, the gravitational coupling g has to be ity with non-zero η report UV-values of 1 + µ consider- ψ strong in order to allow for F < 0. Hence, due to the ably smaller than 4, we conclude that the condition (28) generic ﬂow of g depicted in Figure 4 (right panel) this is not fulﬁlled in the UV. In the IR, where (28) can hold, is the case only in the deep UV. (27) is not fulﬁlled, since η (cid:28) 1 and, thus, δ ≈ 1. ψ C. The sign of F Truncations with zero fermion anomalous dimension η = 0 [31, 39] require very large values of g of the order ψ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: particular,

text: the

text: fermion

text: anomalous

text: dimension

text: at

text: van-

text: The

text: sign

text: of

text: F

text: in

text: the

text: deep

text: UV

text: is

text: not

text: ﬁxed

text: for

text: general

text: ishing

text: momentum

text: η

text: as

text: well

text: as

text: f

text: do

text: not

text: play

text: any

text: rˆole

text: regulators.

text: However,

text: we

text: will

text: show

text: that

text: deﬁnite

text: state-

text: ψ

text: for

text: the

text: sign

text: of

text: left-hand

text: side

text: of

text: since

text: they

text: enter

text: ments

text: about

text: the

text: latter

text: are

text: indeed

text: possible

text: once

text: a

text: regula-

text: quadratically.

text: We

text: have

text: shown

text: that

text: the

text: annihilation

text: of

text: tor

text: is

text: chosen.

text: In

text: the

text: following,

text: we

text: consider

text: x

text: rψ(x)

text: k

text: ﬁxed

text: points

text: due

text: to

text: gravitational

text: interactions

text: in

text: our

text: model

text: x)

text: and

text: x

text: rh(x)

text: x)

text: which

text: k

text: is

text: impossible.

text: The

text: only

text: possible

text: way

text: to

text: destroy

text: the

text: well-

text: allow

text: for

text: analytic

text: ﬂow

text: equations.

text: For

text: these

text: choices,

text: F

text: is

text: deﬁned

text: limits

text: now,

text: is

text: via

text: the

text: change

text: of

text: stability

text: of

text: the

text: given

text: by

text: ﬁxed

text: points.

text: B.

text: Stability

text: of

text: Fixed

text: Points

text: g

text: 2η

text: F

text: h

text: ψ

text: η

text: µ

text: ψ

text: We

text: now

text: turn

text: to

text: our

text: second

text: criterion,

text: the

text: stability

text: of

text: the

text: ﬁxed

text: points

text: in

text: the

text: NJL-model.

text: It

text: is

text: conceivable,

text: that,

text: if

text: the

text: gravitational

text: interaction

text: is

text: strong

text: enough,

text: it

text: might

text: change

text: the

text: attractivity

text: of

text: its

text: In

text: the

text: vicinity

text: where

text: we

text: have

text: moved

text: the

text: anomalous

text: dimensions

text: out

text: of

text: ﬁxed

text: points,

text: the

text: ﬂow

text: can

text: be

text: linearised.

text: Thus,

text: its

text: be-

text: of

text: the

text: loop

text: integrals

text: at

text: momenta

text: at

text: which

text: the

text: integrand

text: haviour

text: is

text: described

text: by

text: critical

text: exponents

text: θ

text: which

text: are

text: is

text: peaked,

text: i.e.

text: This

text: is

text: shown

text: to

text: be

text: a

text: good

text: n

text: the

text: eigenvalues

text: of

text: the

text: wit

text: approximation

text: in

text: Now

text: we

text: investigate,

text: for

text: which

text: ∂λj

text: values

text: of

text: µ,

text: F

text: does

text: become

text: zero,

text: and,

text: thus

text: where

text: F

text: is

text: in

text: principle

text: possible.

text: For

text: the

text: following

text: analysis,

text: ∂β

text: we

text: make

text: use

text: of

text: the

text: bounds

text: of

text: the

text: anomalous

text: dimensions

text: θn

text: Spec

text: ∂λλi

text: F

text: and

text: which

text: are

text: necessary

text: such

text: j

text: n

text: that

text: the

text: corresponding

text: class

text: of

text: regulators,

text: remains

text: well-behaved,

text: for

text: details

text: see

text: We

text: will

text: take

text: η

text: as

text: ψ

text: Hence,

text: from

text: the

text: ﬂow

text: equations

text: together

text: with

text: the

text: momentum-independent

text: and

text: write

text: η

text: η

text: Further-

text: φ

text: φ

text: ﬁxed

text: point

text: values

text: we

text: ﬁnd

text: the

text: critical

text: exponents

text: of

text: more,

text: we

text: assume

text: that

text: g

text: does

text: not

text: take

text: large

text: values

text: g

text: the

text: gravity.

text: simple

text: form

text: along

text: the

text: ﬂow

text: from

text: k

text: to

text: k

text: This

text: assumption

text: is

text: supported

text: by

text: the

text: FRG

text: studies

text: of

text: pure

text: quantum

text: grav-

text: ity

text: and

text: quantum

text: gravity

text: with

text: the

text: given

text: matter

text: content

text: hg

text: hg

text: θ1

text: Nf

text: known

text: to

text: the

text: authors.

text: Summarising

text: the

text: main

text: steps,

text: we

text: solve

text: F

text: for

text: µ,

text: where

text: F

text: given

text: in

text: Employing

text: the

text: approximation

text: for

text: η

text: η

text: and

text: the

text: ψ

text: ψ

text: hg

text: hg

text: regulator

text: constraints

text: ﬁrst

text: leads

text: to

text: θ2

text: −2

text: −1

text: η

text: ψ

text: θ

text: The

text: assumption

text: that

text: g

text: does

text: not

text: become

text: large,

text: implies

text: that

text: η

text: must

text: be

text: very

text: close

text: to

text: the

text: lower

text: limit

text: in

text: We

text: ψ

text: Considering

text: again

text: the

text: limit

text: of

text: vanishing

text: gravitational

text: express

text: this

text: ﬁnding

text: as

text: coupling

text: g

text: thus

text: F

text: and

text: g

text: we

text: get

text: θ

text: θ

text: and

text: θ

text: which

text: we

text: know

text: from

text: the

text: NJL-model

text: without

text: gravity.

text: Note

text: that

text: these

text: ηψ

text: δ

text: quantities

text: are

text: regulator

text: independent

text: in

text: the

text: limit

text: g

text: since

text: the

text: fermion

text: anomalous

text: dimension

text: does

text: not

text: depend

text: which

text: allows

text: to

text: expand

text: the

text: equation

text: for

text: µ

text: in

text: a

text: on

text: the

text: couplings.

text: Hence,

text: this

text: fact

text: is

text: a

text: conse-

text: Taylor

text: series

text: in

text: δ.

text: More

text: details

text: can

text: be

text: found

text: in

text: Appendix

text: quence

text: of

text: the

text: momentum

text: independent

text: interac-

text: The

text: analysis

text: results

text: in

text: a

text: lower

text: bound

text: for

text: µ,

text: namely

text: tions

text: considered

text: here.

text: For

text: g

text: however,

text: the

text: regulator

text: dependent

text: quantities

text: F,

text: h

text: and

text: g

text: enter

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text: Range prediction for tissue mixtures based on dual-energy CT Christian M¨ohler1,2, Patrick Wohlfahrt3,4, Christian − Richter3 6, Steﬀen Greilich1,2 1 German Cancer Research Center (DKFZ), Heidelberg, Germany 2 National Center for Radiation Oncology (NCRO), Heidelberg Institute for Radiation Oncology (HIRO) 3 OncoRay - National Center for Radiation Research in Oncology, Faculty of Medicine and University Hospital Carl Gustav Carus, Technische Universit¨at Dresden, Helmholtz-Zentrum Dresden - Rossendorf, Dresden, Germany 4 Helmholtz-Zentrum Dresden - Rossendorf, Institute of Radiooncology, Dresden, Germany 5 Department of Radiation Oncology, Faculty of Medicine and University Hospital Carl Gustav Carus, Technische Universita¨t Dresden, Dresden, Germany 6 German Cancer Consortium (DKTK), Dresden, Germany E-mail: c.moehler@dkfz-heidelberg.de Abstract. The use of dual-energy CT (DECT) potentially decreases range uncertainties in proton and ion therapy treatment planning via determination of the involved physical target quantities. For eventual clinical application, the correct treatment of tissue mixtures and heterogeneities is an essential feature, as they naturally occur within a patient’s CT. Here, we present how existing methods for DECT- based ion-range prediction can be modiﬁed in order to incorporate proper mixing behavior on several structural levels. Our approach is based on the factorization of the stopping-power ratio into the relative electron density and the relative stopping number. The latter is conﬁned for tissue between about 0.95 and 1.02 at a therapeutic beam energy of 200 MeV/u and depends on the I-value. We show that convenient mixing and averaging properties arise by relating the relative stopping number to the relative cross section obtained by DECT. From this, a maximum uncertainty of the stopping-power ratio prediction below 1% is suggested for arbitrary mixtures of human body tissues. Keywords: proton and ion radiation therapy, treatment planning, computed tomography, volume averaging

text: Range

label: title

text: prediction

text: for

text: tissue

text: mixtures

text: based

text: on

text: dual-energy

text: CT

text: Christian

label: author

text: Patrick

text: Christian

text: Steﬀen

text: German

text: Cancer

text: Research

text: Center

text: (DKFZ),

text: Heidelberg,

text: Germany

text: National

text: Center

text: for

text: Radiation

text: Oncology

text: (NCRO),

text: Heidelberg

text: Institute

text: for

text: Radiation

text: Oncology

text: (HIRO)

text: OncoRay

text: National

text: Center

text: for

text: Radiation

text: Research

text: in

text: Oncology,

text: Faculty

text: of

text: Medicine

text: and

text: University

text: Hospital

text: Carl

text: Gustav

text: Carus,

text: Technische

text: Universit¨at

text: Dresden,

text: Helmholtz-Zentrum

text: Dresden

text: Rossendorf,

text: Dresden,

text: Germany

text: Helmholtz-Zentrum

text: Dresden

text: Rossendorf,

text: Institute

text: of

text: Radiooncology,

text: Dresden,

text: Germany

text: Department

text: of

text: Radiation

text: Oncology,

text: Faculty

text: of

text: Medicine

text: and

text: University

text: Hospital

text: Carl

text: Gustav

text: Carus,

text: Technische

text: Universita¨t

text: Dresden,

text: Dresden,

text: Germany

text: German

text: Cancer

text: Consortium

text: (DKTK),

text: Dresden,

text: Germany

text: E-mail:

text: c.moehler@dkfz-heidelberg.de

text: Abstract.

text: The

text: use

text: of

text: dual-energy

text: CT

text: (DECT)

text: potentially

text: decreases

text: range

text: uncertainties

text: in

text: proton

text: and

text: ion

text: therapy

text: treatment

text: planning

text: via

text: determination

text: of

text: the

text: involved

text: physical

text: target

text: quantities.

text: For

text: eventual

text: clinical

text: application,

text: the

text: correct

text: treatment

text: of

text: tissue

text: mixtures

text: and

text: heterogeneities

text: is

text: an

text: essential

text: feature,

text: as

text: they

text: naturally

text: occur

text: within

text: a

text: patient’s

text: CT.

text: Here,

text: we

text: present

text: how

text: existing

text: methods

text: for

text: DECT-

text: based

text: ion-range

text: prediction

text: can

text: be

text: modiﬁed

text: in

text: order

text: to

text: incorporate

text: proper

text: mixing

text: behavior

text: on

text: several

text: structural

text: levels.

text: Our

text: approach

text: is

text: based

text: on

text: the

text: factorization

text: of

text: the

text: stopping-power

text: ratio

text: into

text: the

text: relative

text: electron

text: density

text: and

text: the

text: relative

text: stopping

text: number.

text: The

text: latter

text: is

text: conﬁned

text: for

text: tissue

text: between

text: about

text: and

text: at

text: a

text: therapeutic

text: beam

text: energy

text: of

text: MeV/u

text: and

text: depends

text: on

text: the

text: I-value.

text: We

text: show

text: that

text: convenient

text: mixing

text: and

text: averaging

text: properties

text: arise

text: by

text: relating

text: the

text: relative

text: stopping

text: number

text: to

text: the

text: relative

text: cross

text: section

text: obtained

text: by

text: DECT.

text: From

text: this,

text: a

text: maximum

text: uncertainty

text: of

text: the

text: stopping-power

text: ratio

text: prediction

text: below

text: is

text: suggested

text: for

text: arbitrary

text: mixtures

text: of

text: human

text: body

text: tissues.

text: Keywords:

text: proton

text: and

text: ion

text: radiation

text: therapy,

text: treatment

text: planning,

text: computed

text: tomography,

text: volume

text: averaging


text: Range prediction for tissue mixtures based on dual-energy CT 3 into the electron density n (referred to as ρ in Hu¨nemohr et al. (2014a)) and the e photon absorption cross section per electron σ. In dimensionless quantities relative to water, indicated by a hat on the variables’ symbols, this equation reads µ(cid:98) = n(cid:98)σ(cid:98) . (1) Via the common deﬁnition of CT numbers, ξ = (µ(cid:98) − 1) · 1000, DECT provides two spectral-weighted relative attenuation coeﬃcients, associated to the two diﬀerent X-ray voltages used (typically 80 or 100 kVp and 140 or 150 kVp, respectively). The attenuation sum rule for chemical compounds or volumetric mixtures (cid:88) µ(cid:98) = µ(cid:98) , (2) i i and (1) yield the relation (cid:88) σ(cid:98) = ν σ(cid:98) (3) i i i for the relative cross sections, where ν are the electron-density fractions of the (cid:80)i constituents deﬁned as ν = n(cid:98) /n(cid:98) with n(cid:98) = n(cid:98) . The following properties arise from i i i i this deﬁnition: (cid:88) 0 < ν < 1 ∀i and ν = 1 . (4) i i i Based on the Bethe formula, the ion stopping-power ratio can be written as L(I, β) S(cid:98) = n(cid:98) =: n(cid:98)L(cid:98) . (5) L(I , β) w with the beam’s relativistic velocity, β, and the I-values of the medium, I, and of water, I . Neglecting shell, density, Barkas and Bloch corrections, we write the w stopping number, L, as 2m c2β2 L(I, β) = ln e − β2 − ln I . (6) 1 − β2 Based on Bragg’s additivity rule for stopping powers (Bragg & Kleemann 1905), (cid:88) S(cid:98) = S(cid:98) , (7) i i and (5), the following holds for compounds or volumetric mixtures in analogy to (3): (cid:88) L(cid:98) = ν L(cid:98) . (8) i i i ##LTLine## ##LTLine##

text: Range

text: prediction

text: for

text: tissue

text: mixtures

text: based

text: on

text: dual-energy

text: CT

text: into

text: the

text: electron

text: density

text: n

text: (referred

text: to

text: as

text: ρ

text: in

text: Hu¨nemohr

text: et

text: al.

text: and

text: the

text: e

text: photon

text: absorption

text: cross

text: section

text: per

text: electron

text: σ.

text: In

text: dimensionless

text: quantities

text: relative

text: to

text: water,

text: indicated

text: by

text: a

text: hat

text: on

text: the

text: variables’

text: symbols,

text: this

text: equation

text: reads

text: Via

text: the

text: common

text: deﬁnition

text: of

text: CT

text: numbers,

text: ξ

text: DECT

text: provides

text: two

text: spectral-weighted

text: relative

text: attenuation

text: coeﬃcients,

text: associated

text: to

text: the

text: two

text: diﬀerent

text: X-ray

text: voltages

text: used

text: (typically

text: or

text: kVp

text: and

text: or

text: kVp,

text: respectively).

text: The

text: attenuation

text: sum

text: rule

text: for

text: chemical

text: compounds

text: or

text: volumetric

text: mixtures

text: i

text: i

text: and

text: yield

text: the

text: relation

text: ν

text: i

text: i

text: i

text: for

text: the

text: relative

text: cross

text: sections,

text: where

text: ν

text: are

text: the

text: electron-density

text: fractions

text: of

text: the

text: constituents

text: deﬁned

text: as

text: ν

text: with

text: The

text: following

text: properties

text: arise

text: from

text: i

text: i

text: i

text: i

text: this

text: deﬁnition:

text: ν

text: ∀i

text: and

text: ν

text: i

text: i

text: i

text: Based

text: on

text: the

text: Bethe

text: formula,

text: the

text: ion

text: stopping-power

text: ratio

text: can

text: be

text: written

text: as

text: L(I,

text: β)

text: L(I

text: β)

text: w

text: with

text: the

text: beam’s

text: relativistic

text: velocity,

text: β,

text: and

text: the

text: I-values

text: of

text: the

text: medium,

text: I,

text: and

text: of

text: water,

text: I

text: Neglecting

text: shell,

text: density,

text: Barkas

text: and

text: Bloch

text: corrections,

text: we

text: write

text: the

text: w

text: stopping

text: number,

text: L,

text: as

text: L(I,

text: β)

text: ln

text: e

text: β2

text: ln

text: I

text: β2

text: Based

text: on

text: Bragg’s

text: additivity

text: rule

text: for

text: stopping

text: powers

text: (Bragg

text: Kleemann

text: i

text: i

text: and

text: the

text: following

text: holds

text: for

text: compounds

text: or

text: volumetric

text: mixtures

text: in

text: analogy

text: to

text: ν

text: i

text: i

text: i

text: ##LTLine##

text: ##LTLine##