text: Log10(Ph) Log10[P11] 100 1014 1 1011 0.01 108 10-4 10-6 105 Λχ 2 4 6 8 10 -k 500 1000 5000 104 H P − Figure 2. Left: Numerical plot of MHP44 T11Pas a function o−f k τ∗(cid:39) for (top to bottom) Λχ = 30 H, 20 H, 10 H. Right: The amplitude of 11 at its peak, k τ∗ 3, as a function of Λ /H. T χ The solid line corresponds to the ﬁt P11(−k τ∗ = 3) = 2.5 × 10−6 Λ5χ , the red bullets correspond to T M4 H P numerical evaluation of the integral (3.33). Log10(Ph) Log1100[27Re[P01]] 100 10 105 1 1000 0.100 0.010 10 0.001 (cid:8) (cid:9) Λχ 2 4 6 8 10 -kτ0 10 50 100 500 1000H (cid:8) (cid:9) P − Figure 3. Left: Numerical plot of MHP44 2 Re T01 as Pa function of k τ∗−for (t(cid:39)op to bottom) Λχ = 30 H, 20 H, 10 H. Right: The amplitude of MP4 2 Re 01 at its peak, k τ∗ 2, as a function of H4 T Λ /H. The solid line corresponds to the analytical ﬁt P11(−k τ∗ = 2) = 4×10−3 Λ3χ H , the red bullets χ T M4 P correspond to numerical evaluation of the integral (3.35). (cid:90) (cid:90) (cid:0) (cid:8) (cid:9)(cid:1) (cid:48) 2 π(cid:2)2 × P − 1 (cid:48) d3q d3q (cid:3) δ(k + k ) 2 Re 01 = dτ k3 1 2 T M2 (2π)3 (cid:2) P (cid:3) × (cid:48) (cid:104) (cid:48) − − (cid:48) (cid:48) (cid:105) (cid:48) (cid:104) − − (cid:48) (cid:48) (cid:48) (cid:105) G (cid:48)(τ, τ ) h(0)(k, τ) h(0)(k q q , τ ) + G (τ, τ ) h(0)(k q q , τ ) h(0)(k , τ) k ij ij k ij ij × (cid:48) (cid:48) (cid:48) (cid:48) (cid:48) · (cid:48) (cid:48) (cid:48) (cid:48) χ (q, τ ) χ (q , τ ) + (q q )χ(q, τ ) χ(q , τ ) , (3.35) where the graviton correlator is given by (cid:18) (cid:19) (cid:48) − (cid:48) (cid:104) (cid:48) (cid:48) (cid:105) 4 δ(3)(k + k ) i(τ τ ) 1 − − (cid:48) h(0)(k, τ) h(0)(k , τ ) = (cid:48) 1 + (cid:48) + (cid:48) e ik (τ τ ) . ij ij a(τ) a(τ ) M2 k k τ τ k2 τ τ P (3.36) → P Taking τ 0, we obtain an expression for 01 that can be integrated numerically for T various values of Λ /H. Spectra for Λ = 10 H, 20 H and 30 H are given in the left χ χ − (cid:39) panel of Figure 3. As we see, the spectra have a peak at k τ∗ 2. In the right panel – 12 – ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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## ##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## ##LTLine## ##LTLine##

text: Λχ

text: -k

text: H

text: P

text: Figure

text: Left:

text: Numerical

text: plot

text: of

text: a

text: function

text: o−f

text: k

text: for

text: (top

text: to

text: bottom)

text: Λχ

text: H,

text: H,

text: H.

text: Right:

text: The

text: amplitude

text: of

text: at

text: its

text: peak,

text: k

text: τ∗

text: as

text: a

text: function

text: of

text: Λ

text: /H.

text: T

text: χ

text: The

text: solid

text: line

text: corresponds

text: to

text: the

text: ﬁt

text: τ∗

text: Λ5χ

text: the

text: red

text: bullets

text: correspond

text: to

text: T

text: H

text: P

text: numerical

text: evaluation

text: of

text: the

text: integral

text: Λχ

text: P

text: Figure

text: Left:

text: Numerical

text: plot

text: of

text: Re

text: as

text: Pa

text: function

text: of

text: k

text: τ∗−for

text: to

text: bottom)

text: Λχ

text: H,

text: H,

text: H.

text: Right:

text: The

text: amplitude

text: of

text: Re

text: at

text: its

text: peak,

text: k

text: τ∗

text: as

text: a

text: function

text: of

text: T

text: Λ

text: /H.

text: The

text: solid

text: line

text: corresponds

text: to

text: the

text: analytical

text: ﬁt

text: τ∗

text: Λ3χ

text: H

text: the

text: red

text: bullets

text: χ

text: T

text: P

text: correspond

text: to

text: numerical

text: evaluation

text: of

text: the

text: integral

text: P

text: δ(k

text: k

text: Re

text: dτ

text: T

text: P

text: G

text: τ

text: τ)

text: q

text: q

text: τ

text: G

text: (τ,

text: τ

text: q

text: q

text: τ

text: τ)

text: k

text: ij

text: ij

text: k

text: ij

text: ij

text: χ

text: (q,

text: τ

text: χ

text: (q

text: τ

text: (q

text: q

text: )χ(q,

text: τ

text: χ(q

text: τ

text: where

text: the

text: graviton

text: correlator

text: is

text: given

text: by

text: k

text: i(τ

text: τ

text: τ)

text: τ

text: e

text: ik

text: (τ

text: τ

text: ij

text: ij

text: a(τ)

text: a(τ

text: k

text: k

text: τ

text: τ

text: τ

text: τ

text: P

text: P

text: Taking

text: τ

text: we

text: obtain

text: an

text: expression

text: for

text: that

text: can

text: be

text: integrated

text: numerically

text: for

text: T

text: various

text: values

text: of

text: Λ

text: /H.

text: Spectra

text: for

text: Λ

text: H,

text: H

text: and

text: H

text: are

text: given

text: in

text: the

text: left

text: χ

text: χ

text: panel

text: of

text: Figure

text: As

text: we

text: see,

text: the

text: spectra

text: have

text: a

text: peak

text: at

text: k

text: τ∗

text: In

text: the

text: right

text: panel

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: µ (cid:48) ϕ¨ + 3 H ϕ˙ + σ2 + V (ϕ) = 0 , 0 0 (cid:18) 2 0 (cid:19) σ¨ + 3 H σ˙ + µ ϕ σ + λ σ3 = 0 , 0 0 0 0 0 1 ϕ˙2 σ˙ 2 µ λ H2 = 0 + 0 + ϕ σ2 + σ4 + V (ϕ) . (4.3) 3 M2 2 2 2 0 0 4 0 P We parametrize the potential near ϕ = 0 as √ (cid:39) − η V (ϕ) 3H¯ 2M2 3 2(cid:15)H¯ 2M ϕ + 3 H¯ 2 ϕ2 , (4.4) P P 2 where (cid:15) and η are the (constant) slow roll parameters. Then, performing the following redeﬁnitions (cid:115) H ϕ λ H˜ = , φ˜ = 0 , σ˜ = σ , H¯ M µM P P ≡ µ M ≡ µ2 P g , g , (4.5) σ H¯ 2 φ λ H¯ 2 we can rewrite the background equations as √ (cid:48)(cid:48) (cid:48) g − φ˜ + 3 H˜ φ˜ + φ σ˜2 3 2 (cid:15) + 3 η φ˜ = 0 , (cid:32) 2 (cid:33) (cid:48)(cid:48) (cid:48) σ˜ + 3 H˜ σ˜ + g φ˜σ˜ + g σ˜3 = 0 , σ σ (cid:48) (cid:48) √ 1 φ˜ 2 g σ˜ 2 g g − 3 H˜ 2 = 1 + + φ + φ φ˜σ˜2 + φ σ˜4 2 (cid:15) 3 φ˜ + η φ˜2 , (4.6) 3 2 g 2 2 4 2 σ where a prime denotes a derivative with respect to H¯ t. Then the conditions √ (cid:28) | | (cid:29) g 6 η , g 2(cid:15) 1 , (cid:113) (4.7) φ σ √ (cid:39) − (cid:39) − (cid:39) are suﬃcient to guarantee that φ˜ 2 (cid:15) H¯ (t t∗), σ˜ φ˜, and H˜ 1 provide good approximations to the√actual solutions for our system for a fe√w efoldings around t = t∗. The (cid:29) equality g = Λ3/(2 2 (cid:15) H¯ 3) implies that the condition g 2(cid:15) 1 is identically veriﬁed σ σ (cid:29) σ since we are already assuming Λ H. σ Therefore we conclude that the only new condition required for the background dynam- (cid:28) | | ics is just g 6 η , which is equivalent to φ µ2 (cid:28) | | 6 η . (4.8) λ H2 – 14 – ##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: µ

label: equation

text: ϕ¨

text: H

text: ϕ˙

text: σ2

text: V

text: (ϕ)

text: σ¨

text: H

text: σ˙

text: µ

text: ϕ

text: σ

text: λ

text: σ3

text: ϕ˙2

text: σ˙

text: µ

text: λ

text: ϕ

text: σ2

text: σ4

text: V

text: (ϕ)

text: P

text: We

label: paragraph

text: parametrize

text: the

text: potential

text: near

text: ϕ

text: as

text: η

text: V

text: (ϕ)

text: ϕ

text: H¯

text: ϕ2

text: P

text: P

text: where

text: and

text: η

text: are

text: the

text: (constant)

text: slow

text: roll

text: parameters.

text: Then,

text: performing

text: the

text: following

text: redeﬁnitions

text: H

text: ϕ

text: λ

text: H˜

text: φ˜

text: σ˜

text: σ

text: H¯

text: M

text: µM

text: P

text: P

text: µ

text: M

text: µ2

text: P

text: g

text: g

text: σ

text: H¯

text: φ

text: λ

text: H¯

text: we

text: can

text: rewrite

text: the

text: background

text: equations

text: as

text: g

text: φ˜

text: H˜

text: φ˜

text: φ

text: σ˜2

text: η

text: φ˜

text: σ˜

text: H˜

text: σ˜

text: g

text: φ˜σ˜

text: g

text: σ˜3

text: σ

text: σ

text: φ˜

text: g

text: σ˜

text: g

text: g

text: H˜

text: φ

text: φ

text: φ˜σ˜2

text: φ

text: σ˜4

text: φ˜

text: η

text: φ˜2

text: g

text: σ

text: where

text: a

text: prime

text: denotes

text: a

text: derivative

text: with

text: respect

text: to

text: H¯

text: t.

text: Then

text: the

text: conditions

text: g

text: η

text: g

text: φ

text: σ

text: are

text: suﬃcient

text: to

text: guarantee

text: that

text: φ˜

text: H¯

text: (t

text: t∗),

text: σ˜

text: φ˜,

text: and

text: H˜

text: provide

text: good

text: approximations

text: to

text: the√actual

text: solutions

text: for

text: our

text: system

text: for

text: a

text: fe√w

text: efoldings

text: around

text: t

text: t∗.

text: The

text: equality

text: g

text: H¯

text: implies

text: that

text: the

text: condition

text: g

text: is

text: identically

text: veriﬁed

text: σ

text: σ

text: σ

text: since

text: we

text: are

text: already

text: assuming

text: Λ

text: H.

text: σ

text: Therefore

text: we

text: conclude

text: that

text: the

text: only

text: new

text: condition

text: required

text: for

text: the

text: background

text: dynam-

text: ics

text: is

text: just

text: g

text: η

text: which

text: is

text: equivalent

text: to

text: φ

text: µ2

text: η

text: λ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4.6 Energy density in the ﬂuctuations of χ By an argument analogous to that of the previous subsection we require the energy in the χ particles to be smaller than the kinetic energy in the zero mode of the σ ﬁeld. (cid:90) (cid:20) (cid:18) (cid:19)(cid:21) Inserting eqs. (3.17) and (3.18) into the expression for the energy in modes of χ, (cid:104) (cid:105) 1 d3p (cid:48) ∗(cid:48) ∗ − 2 ρ = χ χ + χ χ p2 , (4.19) χ 2 a4 (2π)3 p p p p τ2 we obtain an expression that is ultraviolet divergent. To make it ﬁnite we subtract oﬀ the energy in the mo(cid:90)de funct(cid:20)ions computed for Λ = 0, (cid:18) (cid:19)(cid:21) (cid:0) (cid:1) χ(cid:0) (cid:1) (cid:104) (cid:105) → 1 d3p (cid:48) ∗(cid:48) − (cid:48) ∗(cid:48) ∗ − ∗ − 2 ρ χ χ χ˜ χ˜ + χ χ χ˜ χ˜ p2 , (4.20) χ 2 a4 (2π)3 p p p p p p p p τ2 where the functions χ and χ˜ are given in eqs. (3.17) and (3.30) above. Numerical p p integration then gives the result (cid:104) (cid:105) × − a4∗ ρ = 8 10 4Λ4 , (4.21) χ χ a4 that is maximal when a = a∗. (cid:113) Since quanta of χ are produced by the rolling of the ﬁeld σ, energy conservation requires (cid:104) (cid:105) (cid:28) ρ 1σ˙ (t )2 where t is the time at which the production of most of the quanta of χ 2 prod prod ∼ − → χ occurs. We note that, for t < t∗, σ(t) µ ϕ˙∗ (t∗ t), so that σ˙ is divergent as t t∗. λ (cid:39) O − However, the production happens at a typical time of the order t t∗ + (Λ 1). As a (cid:104) (cid:105) (cid:28) O − ∼ χ consequence, we will impose ρ 1σ˙ (t∗ + (Λ 1))2 Λ4 /h2, leading to the constraint (cid:28) χ 2 (cid:46)χ χ h 30 that is always satisﬁed since we require h 1 by perturbativity. 4.7 Eﬀect of the ﬂuctuations of σ on the metric perturbations We next consider how the ﬂuctuations in σ will aﬀect the ﬂuctuations in ϕ. In particular, we want to make sure that those induced ﬂuctuations in ϕ are small compared to the scalar perturbations measured in the CMB. Since the ﬂuctuations of σ are signiﬁcant only after t∗ we set σ = 0 so that we are left with the following equation for the ﬂuctuations in ϕ: 0 (cid:48) (cid:48)(cid:48) a (cid:48) − µ δϕ + 2 δϕ ∆ δϕ + a2 δσ2 = 0 , (4.22) a 2 (cid:90) (cid:90) which we solve as µ (cid:48) (cid:48) (cid:48) d3p (cid:48) − (cid:48) δϕ(k, τ) = dτ a2(τ ) G (τ, τ ) δσ(p, τ ) δσ(k p, τ ) , (4.23) 2 k (2π)3/2 where the Green’s function reads (cid:2) (cid:3) (cid:48) 1 (cid:48) − (cid:48) (cid:48) − − (cid:48) − (cid:48) Gk(τ, τ ) = k3τ(cid:48)2 (1 + k2ττ ) sin(k(τ τ )) + k (τ τ) cos(k(τ τ )) Θ(τ τ ). (4.24) – 17 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: section

text: Energy

text: density

text: in

text: the

text: ﬂuctuations

text: of

text: χ

text: By

text: an

text: argument

text: analogous

text: to

text: that

text: of

text: the

text: previous

text: subsection

text: we

text: require

text: the

text: energy

text: in

text: the

text: χ

text: particles

text: to

text: be

text: smaller

text: than

text: the

text: kinetic

text: energy

text: in

text: the

text: zero

text: mode

text: of

text: the

text: σ

text: ﬁeld.

text: Inserting

text: eqs.

text: and

text: into

text: the

text: expression

text: for

text: the

text: energy

text: in

text: modes

text: of

text: χ,

text: ρ

text: χ

text: χ

text: χ

text: χ

text: χ

text: p

text: p

text: p

text: p

text: τ2

text: we

text: obtain

text: an

text: expression

text: that

text: is

text: ultraviolet

text: divergent.

text: To

text: make

text: it

text: ﬁnite

text: we

text: subtract

text: oﬀ

text: the

text: energy

text: in

text: the

text: computed

text: for

text: Λ

text: ρ

text: χ

text: χ

text: χ˜

text: χ˜

text: χ

text: χ

text: χ˜

text: χ˜

text: χ

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: τ2

text: where

text: the

text: functions

text: χ

text: and

text: χ˜

text: are

text: given

text: in

text: eqs.

text: and

text: above.

text: Numerical

text: p

text: p

text: integration

text: then

text: gives

text: the

text: result

text: ρ

text: χ

text: χ

text: that

text: is

text: maximal

text: when

text: a

text: a∗.

text: Since

text: quanta

text: of

text: χ

text: are

text: produced

text: by

text: the

text: rolling

text: of

text: the

text: ﬁeld

text: σ,

text: energy

text: conservation

text: requires

text: ρ

text: 1σ˙

text: (t

text: where

text: t

text: is

text: the

text: time

text: at

text: which

text: the

text: production

text: of

text: most

text: of

text: the

text: quanta

text: of

text: χ

text: prod

text: prod

text: χ

text: occurs.

text: We

text: note

text: that,

text: for

text: t

text: t∗,

text: σ(t)

text: µ

text: ϕ˙∗

text: (t∗

text: t),

text: so

text: that

text: σ˙

text: is

text: divergent

text: as

text: t

text: t∗.

text: λ

text: O

text: However,

text: the

text: production

text: happens

text: at

text: a

text: typical

text: time

text: of

text: the

text: order

text: t

text: t∗

text: (Λ

text: As

text: a

text: O

text: χ

text: consequence,

text: we

text: will

text: impose

text: ρ

text: 1σ˙

text: (t∗

text: (Λ

text: Λ4

text: leading

text: to

text: the

text: constraint

text: χ

text: χ

text: h

text: that

text: is

text: always

text: satisﬁed

text: since

text: we

text: require

text: h

text: by

text: perturbativity.

text: Eﬀect

text: of

text: the

text: ﬂuctuations

text: of

text: σ

text: on

text: the

text: metric

text: perturbations

text: We

text: next

text: consider

text: how

text: the

text: ﬂuctuations

text: in

text: σ

text: will

text: aﬀect

text: the

text: ﬂuctuations

text: in

text: ϕ.

text: In

text: particular,

text: we

text: want

text: to

text: make

text: sure

text: that

text: those

text: induced

text: ﬂuctuations

text: in

text: ϕ

text: are

text: small

text: compared

text: to

text: the

text: scalar

text: perturbations

text: measured

text: in

text: the

text: CMB.

text: Since

text: the

text: ﬂuctuations

text: of

text: σ

text: are

text: signiﬁcant

text: only

text: after

text: t∗

text: we

text: set

text: σ

text: so

text: that

text: we

text: are

text: left

text: with

text: the

text: following

text: equation

text: for

text: the

text: ﬂuctuations

text: in

text: ϕ:

text: a

text: µ

text: δϕ

text: δϕ

text: δϕ

text: δσ2

text: a

text: which

text: we

text: solve

text: as

text: µ

text: δϕ(k,

text: τ)

text: dτ

text: G

text: (τ,

text: τ

text: δσ(p,

text: τ

text: δσ(k

text: p,

text: τ

text: k

text: where

text: the

text: Green’s

text: function

text: reads

text: Gk(τ,

text: τ

text: k2ττ

text: sin(k(τ

text: τ

text: k

text: (τ

text: τ)

text: cos(k(τ

text: τ

text: Θ(τ

text: τ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 0.000015 0.00001 5.×10-6 0 5 10 15 20 − Figure 5. Plot of the function f ( kτ∗) deﬁned in eq. (4.30). δϕ   (cid:90)  (cid:113)  (cid:0) (cid:1) y − 2 (cid:39) .51 dx sin(x) + x cos(x) f(y) , (4.30) 64π5y3 x ln y 0 x (cid:39) × − is plotted in Figure 5 and is maximized at y 3 where it evaluates to 1.7 10 5. To sum up, the requirement that the metric perturbations induced by the interactions between the inﬂaton and the ﬁeld σ do not exceed the measured amplitude of the scalar power spectrum leads to the constraint (cid:18) (cid:19) ϕ˙2∗ P (cid:39) H 2 (cid:29) P ⇒ × − ϕ˙2∗ (cid:29) − sourced(k) 25 10 10 µ2f( kτ∗) , (4.31) H2 ζ 2π δϕ H2 or, equivalently, µ2 (cid:28) × − 1.5 10 4 . (4.32) 2(cid:15) M2 P Before concluding this Section, we note that the sourced scalar perturbations will obey non-gaussian statistics and would be in principle subject to the strong constraints from the Planck satellite on the amplitude of equilateral bispectra [35]. However, those strong constraints hold when the nongaussian component of the scalar perturbations has a (quasi) scale invariant component. Since the contribution (4.29) is strongly scale dependent, we do not expect the non-observation of equilateral nongaussianities to constrain the parameter space of the model more strongly than eq. (4.32), similarly to what was found in [16]. 5 How large of a spectrum for induced PGWs can be generated? The spectrum of produced gravitational waves is proportional to Λ5 , and in this Section χ we estimate how large Λ can be once the constraints of the previous section are enforced. χ The constraints found in the previous section can be summarized as follows – 19 – ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##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##

font: default

label: caption

text: Figure

text: Plot

text: of

text: the

text: function

text: f

text: kτ∗)

text: deﬁned

text: in

text: eq.

text: δϕ

text: y

text: dx

text: sin(x)

text: x

text: cos(x)

text: f(y)

text: x

text: ln

text: y

text: x

text: is

text: plotted

text: in

text: Figure

text: and

text: is

text: maximized

text: at

text: y

text: where

text: it

text: evaluates

text: to

text: To

text: sum

text: up,

text: the

text: requirement

text: that

text: the

text: metric

text: perturbations

text: induced

text: by

text: the

text: interactions

text: between

text: the

text: inﬂaton

text: and

text: the

text: ﬁeld

text: σ

text: do

text: not

text: exceed

text: the

text: measured

text: amplitude

text: of

text: the

text: scalar

text: power

text: spectrum

text: leads

text: to

text: the

text: constraint

text: ϕ˙2∗

text: P

text: H

text: P

text: ϕ˙2∗

text: sourced(k)

text: kτ∗)

text: ζ

text: 2π

text: δϕ

text: or,

text: equivalently,

text: µ2

text: P

text: Before

text: concluding

text: this

text: Section,

text: we

text: note

text: that

text: the

text: sourced

text: scalar

text: perturbations

text: will

text: obey

text: non-gaussian

text: statistics

text: and

text: would

text: be

text: in

text: principle

text: subject

text: to

text: the

text: strong

text: constraints

text: from

text: the

text: Planck

text: satellite

text: on

text: the

text: amplitude

text: of

text: equilateral

text: bispectra

text: However,

text: those

text: strong

text: constraints

text: hold

text: when

text: the

text: nongaussian

text: component

text: of

text: the

text: scalar

text: perturbations

text: has

text: a

text: (quasi)

text: scale

text: invariant

text: component.

text: Since

text: the

text: contribution

text: is

text: strongly

text: scale

text: dependent,

text: we

text: do

text: not

text: expect

text: the

text: non-observation

text: of

text: equilateral

text: nongaussianities

text: to

text: constrain

text: the

text: parameter

text: space

text: of

text: the

text: model

text: more

text: strongly

text: than

text: eq.

text: similarly

text: to

text: what

text: was

text: found

text: in

text: How

text: large

text: of

text: a

text: spectrum

text: for

text: induced

text: PGWs

text: can

text: be

text: generated?

text: The

text: spectrum

text: of

text: produced

text: gravitational

text: waves

text: is

text: proportional

text: to

text: Λ5

text: and

text: in

text: this

text: Section

text: χ

text: we

text: estimate

text: how

text: large

text: Λ

text: can

text: be

text: once

text: the

text: constraints

text: of

text: the

text: previous

text: section

text: are

text: enforced.

text: χ

text: The

text: constraints

text: found

text: in

text: the

text: previous

text: section

text: can

text: be

text: summarized

text: as

text: follows

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

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: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ (cid:29) (i) 2µ 2√(cid:15) M H2 Subsection 4.2 , P h2 µ 2(cid:15) M (cid:29) (ii) P H2 Subsection 4.2 , λ µ2 (cid:28) | | (iii) 6 η Subsection 4.3 , λ H2 µ4 (cid:28) (iv) 2 (cid:15) H2 M2 Subsection 4.4 , 25λ3 P √ µ3 (cid:28) × (v) 1.7 105 Subsection 4.5 , 2(cid:15) H2 M P µ2 (cid:28) × − (vi) 1.5 10 4 Subsection 4.7 , (5.1) 2(cid:15) M2 P besides the perturbativity requirements h, λ < 1. 5.1 Detectability at CMB scales Let us ﬁrst consider the maximal amplitude of the sourced gravitational waves at CMB scales, where the dynamics of the inﬂaton is constrained by CMB observations. First, we (cid:39) trade M for H and (cid:15) using COBE normalization 2(cid:15) 107 H2/M2 : P P (cid:29) × − (i) µ 1.7 10 4 H , h2 (cid:29) × − (ii) µ 1.7 10 4 H , λ µ2 (cid:28) | | (iii) 6 η H2 , λ µ4 (cid:28) × (iv) 3 108 H4 , λ3 (cid:28) × (v) µ3 5 108 H3 , (cid:28) × (vi) µ2 1.5 103 H2 . (5.2) We now note, ﬁrst, that h appears only in condition (ii). In order to maximize the volume of our parameter space while remaining within the perturbative regime we set from now on h = 1. Moreover, we note that, once the perturbativity requirement λ < 1 is imposed, conditions (i), (iii), (iv), (v) and (vi) reduce simply to conditions (i) and (iii). Therefore we are left just with (cid:29) × − (i) µ 1.7 10 4 H , µ2 (cid:28) | | (iii) 6 η H2 . (5.3) λ We remember that we are seeking to maximize Λ /H, which is proportional to (µ/(λ H))1/3. χ Trading µ for Λ in the equations above by using χ – 20 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: (i)

text: 2µ

text: M

text: Subsection

text: P

text: µ

text: M

text: (ii)

text: P

text: Subsection

text: λ

text: µ2

text: (iii)

text: η

text: Subsection

text: λ

text: µ4

text: (iv)

text: Subsection

text: P

text: µ3

text: (v)

text: Subsection

text: M

text: P

text: µ2

text: (vi)

text: Subsection

text: P

text: besides

text: the

text: perturbativity

text: requirements

text: h,

text: λ

text: Detectability

text: at

text: CMB

text: scales

text: Let

text: us

text: ﬁrst

text: consider

text: the

text: maximal

text: amplitude

text: of

text: the

text: sourced

text: gravitational

text: waves

text: at

text: CMB

text: scales,

text: where

text: the

text: dynamics

text: of

text: the

text: inﬂaton

text: is

text: constrained

text: by

text: CMB

text: observations.

text: First,

text: we

text: trade

text: M

text: for

text: H

text: and

text: using

text: COBE

text: normalization

text: P

text: P

text: (i)

text: µ

text: H

text: (ii)

text: µ

text: H

text: λ

text: µ2

text: (iii)

text: η

text: λ

text: µ4

text: (iv)

text: λ3

text: (v)

text: µ3

text: (vi)

text: µ2

text: We

text: now

text: note,

text: ﬁrst,

text: that

text: h

text: appears

text: only

text: in

text: condition

text: (ii).

text: In

text: order

text: to

text: maximize

text: the

text: volume

text: of

text: our

text: parameter

text: space

text: while

text: remaining

text: within

text: the

text: perturbative

text: regime

text: we

text: set

text: from

text: now

text: on

text: h

text: Moreover,

text: we

text: note

text: that,

text: once

text: the

text: perturbativity

text: requirement

text: λ

text: is

text: imposed,

text: conditions

text: (i),

text: (iii),

text: (iv),

text: (v)

text: and

text: (vi)

text: reduce

text: simply

text: to

text: conditions

text: (i)

text: and

text: (iii).

text: Therefore

text: we

text: are

text: left

text: just

text: with

text: (i)

text: µ

text: H

text: µ2

text: (iii)

text: η

text: λ

text: We

text: remember

text: that

text: we

text: are

text: seeking

text: to

text: maximize

text: Λ

text: /H,

text: which

text: is

text: proportional

text: to

text: (µ/(λ

text: χ

text: Trading

text: µ

text: for

text: Λ

text: in

text: the

text: equations

text: above

text: by

text: using

text: χ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − − where M = (8πG) 1/2 is the reduced Planck mass, Π lm = ΠlΠm 1Π Πlm is the P − ij i j 2 ij transverse, traceless projector, Π = δ ∂ ∂ /∆, and a prime denotes derivatives with ij ij i j respect to conformal time τ . As we have discussed in the Introduction, our goal is to consider a scenario where the mass of a scalar ﬁeld χ goes from a nonvanishing to a vanishing value during inﬂation. We do so by considering an additional ﬁeld σ which controls the mass of χ and that behaves as an order parameter in a phase transition describing a symmetry restoration. More speciﬁcally, we will consider a system where a ﬁeld σ and the inﬂaton ϕ are subject to a potential of the form L −1 − 1 − µ − λ − = ∂ ϕ∂µϕ ∂ σ∂µσ ϕ σ2 σ4 V (ϕ) , (2.3) ϕσ µ µ 2 2 2 4 where V (ϕ) is some ﬂat potential able to support inﬂation, λ is a dimensionless coupling constant, and µ is a mass dimension-1 coupling constant. The coupling between ϕ and σ − would generally take the form µ (ϕ ϕ∗) σ2, where ϕ∗ is some constant value crossed by the 2 expectation value of ϕ during inﬂation. However, we can always set ϕ∗ = 0 by an appropriate shift of ϕ. Reference [33] also studied a system where a ﬁeld analogous to σ evolved towards a vanishing expectation value during inﬂation. In that paper, however, that ﬁeld was not coupled to the inﬂaton, so that its evolution was more slow than in the present work. We will assume without loss of generality that ϕ˙ > 0, so that the term proportional to µ in the Lagrangian (2.3) behaves like a negative mass squared term for σ at early times, (cid:113) triggering symmetry breaking, while at later times it behaves like a positive mass term, enforcing σ = 0. More explicitly, for ϕ < 0 the minimum of the potent(cid:113)ial for σ is σ = min ± − µϕ, while for ϕ > 0, the minimum is σ = 0. We will assume that some earlier λ min − inﬂationary dynamics has chosen one of the two minima, say σ = + µϕ > 0 for the min λ early value of the zero mode of the σ ﬁeld. Let us now introduce a third ﬁeld χ, that will be our source of gravitational waves. The ﬁeld χ interacts with σ through the lagrangian L −1 − h2 = ∂ χ∂µχ σ2χ2, (2.4) χ µ 2 2 where h is a dimensionless coupling constant. If σ tracks the minimum of its potential (we will see in Subsection under which conditions this requirement is satisﬁed) then the (cid:40)(cid:113) mass of χ will be given by − h2 µ ϕ for t < t∗ m = λ (2.5) χ 0 for t > t∗, where t∗ corresponds to the time when ϕ crosses 0. If the inﬂaton evolves under the usual slow-roll conditions then we can model its time evolution around t∗ as (cid:39) − ϕ(t) ϕ˙∗ (t t∗) , (2.6) – 4 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: M

text: is

text: the

text: reduced

text: Planck

text: mass,

text: Π

text: lm

text: ΠlΠm

text: 1Π

text: Πlm

text: is

text: the

text: P

text: ij

text: i

text: j

text: ij

text: transverse,

text: traceless

text: projector,

text: Π

text: δ

text: and

text: a

text: prime

text: denotes

text: derivatives

text: with

text: ij

text: ij

text: i

text: j

text: respect

text: to

text: conformal

text: time

text: τ

text: As

text: we

text: have

text: discussed

text: in

text: the

text: Introduction,

text: our

text: goal

text: is

text: to

text: consider

text: a

text: scenario

text: where

text: the

text: mass

text: of

text: a

text: scalar

text: ﬁeld

text: χ

text: goes

text: from

text: a

text: nonvanishing

text: to

text: a

text: vanishing

text: value

text: during

text: inﬂation.

text: We

text: do

text: so

text: by

text: considering

text: an

text: additional

text: ﬁeld

text: σ

text: which

text: controls

text: the

text: mass

text: of

text: χ

text: and

text: that

text: behaves

text: as

text: an

text: order

text: parameter

text: in

text: a

text: phase

text: transition

text: describing

text: a

text: symmetry

text: restoration.

text: More

text: speciﬁcally,

text: we

text: will

text: consider

text: a

text: system

text: where

text: a

text: ﬁeld

text: σ

text: and

text: the

text: inﬂaton

text: ϕ

text: are

text: subject

text: to

text: a

text: potential

text: of

text: the

text: form

text: L

text: −1

text: µ

text: λ

text: ϕ∂µϕ

text: σ∂µσ

text: ϕ

text: σ2

text: σ4

text: V

text: (ϕ)

text: ϕσ

text: µ

text: µ

text: where

text: V

text: (ϕ)

text: is

text: some

text: ﬂat

text: potential

text: able

text: to

text: support

text: inﬂation,

text: λ

text: is

text: a

text: dimensionless

text: coupling

text: constant,

text: and

text: µ

text: is

text: a

text: mass

text: coupling

text: constant.

text: The

text: coupling

text: between

text: ϕ

text: and

text: σ

text: would

text: generally

text: take

text: the

text: form

text: µ

text: (ϕ

text: ϕ∗)

text: where

text: ϕ∗

text: is

text: some

text: constant

text: value

text: crossed

text: by

text: the

text: expectation

text: value

text: of

text: ϕ

text: during

text: inﬂation.

text: However,

text: we

text: can

text: always

text: set

text: ϕ∗

text: by

text: an

text: appropriate

text: shift

text: of

text: ϕ.

text: Reference

text: also

text: studied

text: a

text: system

text: where

text: a

text: ﬁeld

text: analogous

text: to

text: σ

text: evolved

text: towards

text: a

text: vanishing

text: expectation

text: value

text: during

text: inﬂation.

text: In

text: that

text: paper,

text: however,

text: that

text: ﬁeld

text: was

text: not

text: coupled

text: to

text: the

text: inﬂaton,

text: so

text: that

text: its

text: evolution

text: was

text: more

text: slow

text: than

text: in

text: the

text: present

text: work.

text: We

text: will

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: ϕ˙

text: so

text: that

text: the

text: term

text: proportional

text: to

text: µ

text: in

text: the

text: Lagrangian

text: behaves

text: like

text: a

text: negative

text: mass

text: squared

text: term

text: for

text: σ

text: at

text: early

text: times,

text: triggering

text: symmetry

text: breaking,

text: while

text: at

text: later

text: times

text: it

text: behaves

text: like

text: a

text: positive

text: mass

text: term,

text: enforcing

text: σ

text: More

text: explicitly,

text: for

text: ϕ

text: the

text: minimum

text: of

text: the

text: for

text: σ

text: is

text: σ

text: min

text: µϕ,

text: while

text: for

text: ϕ

text: the

text: minimum

text: is

text: σ

text: We

text: will

text: assume

text: that

text: some

text: earlier

text: λ

text: min

text: inﬂationary

text: dynamics

text: has

text: chosen

text: one

text: of

text: the

text: two

text: minima,

text: say

text: σ

text: µϕ

text: for

text: the

text: min

text: λ

text: early

text: value

text: of

text: the

text: zero

text: mode

text: of

text: the

text: σ

text: ﬁeld.

text: Let

text: us

text: now

text: introduce

text: a

text: third

text: ﬁeld

text: χ,

text: that

text: will

text: be

text: our

text: source

text: of

text: gravitational

text: waves.

text: The

text: ﬁeld

text: χ

text: interacts

text: with

text: σ

text: through

text: the

text: lagrangian

text: L

text: −1

text: χ∂µχ

text: χ

text: µ

text: where

text: h

text: is

text: a

text: dimensionless

text: coupling

text: constant.

text: If

text: σ

text: tracks

text: the

text: minimum

text: of

text: its

text: potential

text: (we

text: will

text: see

text: in

text: Subsection

text: under

text: which

text: conditions

text: this

text: requirement

text: is

text: satisﬁed)

text: then

text: the

text: mass

text: of

text: χ

text: will

text: be

text: given

text: by

text: µ

text: ϕ

text: for

text: t

text: t∗

text: m

text: λ

text: χ

text: for

text: t

text: t∗,

text: where

text: t∗

text: corresponds

text: to

text: the

text: time

text: when

text: ϕ

text: crosses

text: If

text: the

text: inﬂaton

text: evolves

text: under

text: the

text: usual

text: slow-roll

text: conditions

text: then

text: we

text: can

text: model

text: its

text: time

text: evolution

text: around

text: t∗

text: as

text: ϕ(t)

text: ϕ˙∗

text: (t

text: t∗)

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: |β 2 0.100 0.001 - 10 5 - Λ 10 7 *p σ 2 4 6 8 10 a | | Figure 1. The occupation number of σ particles β 2 as a function of the momentum p expressed σ in units of a∗ Λ . σ | | The occupation number β 2 is plotted in ﬁgure 1. We will use the above coeﬃcients σ to determine the eﬀects of the produced σ particles in Subsections and below. 3.2 Production of quanta of χ The analysis of the production of quanta of χ is similar to that of the previous Sub- section, with the additional complication that the χ particles will remain massless (and the superhorizon modes will therefore not be evolving adiabatically) after the event of particle (cid:90) (cid:104) (cid:105) production. We decompose the ﬁeld χ as 1 d3p · ∗ † χˆ(x, τ) = eip x χ (τ) aˆ + χ (τ) aˆ , (3.12) a(τ) (2π)3/2 p p -p -p (cid:20) (cid:21) where the (canonically normalized) mode functions satisfy (cid:48)(cid:48) (cid:48)(cid:48) − a χ + p2 + m2 (τ) a2 χ = 0 , (3.13) p a χ p with m (τ) given by eq. (2.7). We will assume that the parameters of the system are χ | (cid:48)| (cid:28) such that m evolves adiabatically, (a m ) a2 m2 for most of the time, and that the χ χ χ period in which the adiabaticity condition is violated, close to the time when m = 0, is χ much shorter than an Hubble time, which implies the condition (cid:29) Λ H . (3.14) χ We can now determine the mode functions during the nonadiabatic regime by switching from conformal to physical time, neglecting the expansion of the Universe during this epoch, (cid:20) − (cid:21) 1 and introducing the rescaled ﬁeld χ = a 2 χc. Then the equation for χc simpliﬁes to p2 − χ¨ + + Λ3 (t∗ t) χ = 0 , (3.15) c a2∗ χ c whose solution for t < t∗, reducing to the adiabatic vacuum at early times, can be written as – 7 – ##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##

text: |β

text: Λ

text: *p

text: σ

text: a

text: Figure

text: The

text: occupation

text: number

text: of

text: σ

text: particles

text: β

text: as

text: a

text: function

text: of

text: the

text: momentum

text: p

text: expressed

text: σ

text: in

text: units

text: of

text: a∗

text: Λ

text: σ

text: The

text: occupation

text: number

text: β

text: is

text: plotted

text: in

text: ﬁgure

text: We

text: will

text: use

text: the

text: above

text: coeﬃcients

text: σ

text: to

text: determine

text: the

text: eﬀects

text: of

text: the

text: produced

text: σ

text: particles

text: in

text: Subsections

text: and

text: below.

text: Production

text: of

text: quanta

text: of

text: χ

text: The

text: analysis

text: of

text: the

text: production

text: of

text: quanta

text: of

text: χ

text: is

text: similar

text: to

text: that

text: of

text: the

text: previous

text: Sub-

text: section,

text: with

text: the

text: additional

text: complication

text: that

text: the

text: χ

text: particles

text: will

text: remain

text: massless

text: (and

text: the

text: superhorizon

text: modes

text: will

text: therefore

text: not

text: be

text: evolving

text: adiabatically)

text: after

text: the

text: event

text: of

text: particle

text: production.

text: We

text: decompose

text: the

text: ﬁeld

text: χ

text: as

text: χˆ(x,

text: τ)

text: eip

text: x

text: χ

text: (τ)

text: aˆ

text: χ

text: (τ)

text: aˆ

text: a(τ)

text: p

text: p

text: -p

text: -p

text: where

text: the

text: (canonically

text: normalized)

text: mode

text: functions

text: satisfy

text: a

text: χ

text: (τ)

text: χ

text: p

text: a

text: χ

text: p

text: with

text: m

text: (τ)

text: given

text: by

text: eq.

text: We

text: will

text: assume

text: that

text: the

text: parameters

text: of

text: the

text: system

text: are

text: χ

text: such

text: that

text: m

text: evolves

text: adiabatically,

text: (a

text: m

text: for

text: most

text: of

text: the

text: time,

text: and

text: that

text: the

text: χ

text: χ

text: χ

text: period

text: in

text: which

text: the

text: adiabaticity

text: condition

text: is

text: violated,

text: close

text: to

text: the

text: time

text: when

text: m

text: is

text: χ

text: much

text: shorter

text: than

text: an

text: Hubble

text: time,

text: which

text: implies

text: the

text: condition

text: Λ

text: H

text: χ

text: We

text: can

text: now

text: determine

text: the

text: mode

text: functions

text: during

text: the

text: nonadiabatic

text: regime

text: by

text: switching

text: from

text: conformal

text: to

text: physical

text: time,

text: neglecting

text: the

text: expansion

text: of

text: the

text: Universe

text: during

text: this

text: epoch,

text: and

text: introducing

text: the

text: rescaled

text: ﬁeld

text: χ

text: a

text: χc.

text: Then

text: the

text: equation

text: for

text: χc

text: simpliﬁes

text: to

text: χ¨

text: Λ3

text: (t∗

text: t)

text: χ

text: c

text: χ

text: c

text: whose

text: solution

text: for

text: t

text: t∗,

text: reducing

text: to

text: the

text: adiabatic

text: vacuum

text: at

text: early

text: times,

text: can

text: be

text: written

text: as

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: VLA 5.5 GHz observations of the GOODS-North Field 11 Figure 8. The four IR colour-colour plots used to classify the 5.5GHz radio sources in the GOODS-N ﬁeld. In each panel diﬀerent colours are used to identify diﬀerent classes of sources based on the classiﬁcation obtained by combining all the four IR plots: RE-AGN are shown in green, RE-AGN-candidates in cyan, RI-AGN in red, and the remaining sources (SF/hyb galaxies) in blue. Top-Left: IRAC colour-colour plot, the dashed-line wedge shows the AGN selection region from Lacy et al. (2007), while the smaller area enclosed by a solid-line is the revised wedge from Donley et al. (2012), used in this paper. Top-Right: KI colour-colour plot (Messias et al. 2012). − − The region populated by AGN is delimited by the solid line deﬁned as Ks [4.5] > 0 and [4.5] [8.0] > 0. Magnitudes are in the AB photometric system. Bottom-Left: IR colour-colour plot showing log(S8.0/S3.6) versus log(S250/S24). The line separate RE-AGN (above the line) from star-forming or passive galaxies (below the line), according to Kirkpatrick et al. (2012, 2013) Bottom-Right: IR colour-colour plot showing log(S8.0/S3.6) versus log(S100/S24). The line separates RE-AGN (above the line) from star-forming or passive galaxies (below the line), according to Kirkpatrick et al. (2012, 2013). In the two lower plots, empty symbols refer to 100µm and 250µm upper limits. − − galaxies have MIR colours consistent with those expected AGN the sources with K [4.5] > 0 and [4.5] [8.0] > 0, s for red and passive galaxies at z <∼ 1. To allow for possi- where AB magnitudes are used. This method selects the (cid:39) ble higher redshift (z 2) quiescent galaxies (see Fig. 2 largest number of AGN (15 RE-AGN and 11 RE-AGN- in Donley et al. (2012) for the evolutionary tracks of pas- candidates) and delivers the highest level of completeness sive galaxies), we consider all the 15 radio sources with both for the RE-AGN (88 percent, 15/19 sources) and − log(S /S ) < 0.05 and log(S /S ) < 0.25 as radio- the overall RE-AGN and RE-AGN-candidates (81 percent, 5.8 3.6 8.0 4.5 emitting RI-AGN hosted by red passive galaxies. 26/32 sources). However, given that 11 radio sources are The KI criterion (top-right panel in Fig. 8) identiﬁes as identiﬁed as AGN-candidates only by this diagnostic, we MNRAS 000, 000–000 (0000)

text: VLA

text: GHz

text: observations

text: of

text: the

text: GOODS-North

text: Field

text: Figure

text: The

text: four

text: IR

text: colour-colour

text: plots

text: used

text: to

text: classify

text: the

text: radio

text: sources

text: in

text: the

text: GOODS-N

text: ﬁeld.

text: In

text: each

text: panel

text: diﬀerent

text: colours

text: are

text: used

text: to

text: identify

text: diﬀerent

text: classes

text: of

text: sources

text: based

text: on

text: the

text: classiﬁcation

text: obtained

text: by

text: combining

text: all

text: the

text: four

text: IR

text: plots:

text: RE-AGN

text: are

text: shown

text: in

text: green,

text: RE-AGN-candidates

text: in

text: cyan,

text: RI-AGN

text: in

text: red,

text: and

text: the

text: remaining

text: sources

text: (SF/hyb

text: galaxies)

text: in

text: blue.

text: Top-Left:

text: IRAC

text: colour-colour

text: plot,

text: the

text: dashed-line

text: wedge

text: shows

text: the

text: AGN

text: selection

text: region

text: from

text: Lacy

text: et

text: al.

text: while

text: the

text: smaller

text: area

text: enclosed

text: by

text: a

text: solid-line

text: is

text: the

text: revised

text: wedge

text: from

text: Donley

text: et

text: al.

text: used

text: in

text: this

text: paper.

text: Top-Right:

text: KI

text: colour-colour

text: plot

text: (Messias

text: et

text: al.

text: The

text: region

text: populated

text: by

text: AGN

text: is

text: delimited

text: by

text: the

text: solid

text: line

text: deﬁned

text: as

text: Ks

text: and

text: Magnitudes

text: are

text: in

text: the

text: AB

text: photometric

text: system.

text: Bottom-Left:

text: IR

text: colour-colour

text: plot

text: showing

text: versus

text: The

text: line

text: separate

text: RE-AGN

text: (above

text: the

text: line)

text: from

text: star-forming

text: or

text: passive

text: galaxies

text: (below

text: the

text: line),

text: according

text: to

text: Kirkpatrick

text: et

text: al.

text: Bottom-Right:

text: IR

text: colour-colour

text: plot

text: showing

text: versus

text: The

text: line

text: separates

text: RE-AGN

text: (above

text: the

text: line)

text: from

text: star-forming

text: or

text: passive

text: galaxies

text: (below

text: the

text: line),

text: according

text: to

text: Kirkpatrick

text: et

text: al.

text: In

text: the

text: two

text: lower

text: plots,

text: empty

text: symbols

text: refer

text: to

text: and

text: upper

text: limits.

text: galaxies

text: have

text: MIR

text: colours

text: consistent

text: with

text: those

text: expected

text: AGN

text: the

text: sources

text: with

text: K

text: and

text: s

text: for

text: red

text: and

text: passive

text: galaxies

text: at

text: z

text: To

text: allow

text: for

text: possi-

text: where

text: AB

text: magnitudes

text: are

text: used.

text: This

text: method

text: selects

text: the

text: ble

text: higher

text: redshift

text: (z

text: quiescent

text: galaxies

text: (see

text: Fig.

text: largest

text: number

text: of

text: AGN

text: RE-AGN

text: and

text: RE-AGN-

text: in

text: Donley

text: et

text: al.

text: for

text: the

text: evolutionary

text: tracks

text: of

text: pas-

text: candidates)

text: and

text: delivers

text: the

text: highest

text: level

text: of

text: completeness

text: sive

text: galaxies),

text: we

text: consider

text: all

text: the

text: radio

text: sources

text: with

text: both

text: for

text: the

text: RE-AGN

text: percent,

text: sources)

text: and

text: log(S

text: /S

text: and

text: log(S

text: /S

text: as

text: radio-

text: the

text: overall

text: RE-AGN

text: and

text: RE-AGN-candidates

text: percent,

text: emitting

text: RI-AGN

text: hosted

text: by

text: red

text: passive

text: galaxies.

text: sources).

text: However,

text: given

text: that

text: radio

text: sources

text: are

text: The

text: KI

text: criterion

text: (top-right

text: panel

text: in

text: Fig.

text: identiﬁes

text: as

text: identiﬁed

text: as

text: AGN-candidates

text: only

text: by

text: this

text: diagnostic,

text: we

text: MNRAS


text: 14 D. Guidetti et al. ties, from normal, radio-quiet spirals to ultra-luminous IR We ﬁnd ﬁve radio sources at z > 1.5, classiﬁed as galaxies (ULIRGs) (Yun et al. 2001; Garrett 2002; Appleton SF/hyb systems and not strong X-ray emitters, that show a et al. 2004), and is one of the most useful diagnostic tools clear radio-excess (q < 1 for four of them). Even taking 100 in revealing excess radio emission exceeding that expected into account the evolution with redshift of q , all these 5 100 from pure star-formation processes. sources would fall in the radio-excess region (e.g. see Fig.5 in Several studies have replaced the FIR ﬂux with the Del Moro et al. 2013). Finally, there is an IR-excess source monochromatic ﬂux at 24 µm (e.g. Bonzini et al. 2013) or at at z = 2.5: here the IR emission should be dominated by 100 µm (e.g. Del Moro et al. 2013) as a proxy for the FIR an AGN and indeed the sources is classiﬁed as such on the emission. In particular, Del Moro et al. (2013) showed that a basis of both the IR and X-ray diagnostics. It does not show simple cut at q < 1.5 (where q = log(S /S )) a radio excess as typically observed in many RE-AGN. The ∼ 100 100 100µm 1.4GHz selects 80 percent of the radio-excess sources deﬁned us- radio excess sources are shown as black squares in Fig. 9. ing the total FIR. Following Del Moro et al. (2013) we use In Table 3 we list, for each of the IR classes, the number q100 < 1.5 to identify radio-excess sources in our radio se- of X-ray AGN, the number of radio-excess sources and the lected sample. Clearly, such a criterion is simplistic and does number of sources which are both X-ray and radio-excess not take into account the z dependence, such that it may suf- AGN. At this point it is important to recall that we have fer of contamination, especially at high redshift, by strong IR classiﬁed as RE-AGN-candidates those sources selected as sources (i.e. hyper-luminous IR galaxies): ideally, q100 should AGN by only one of the four IR diagnostics. The reason for be compared to the expected tracks for diﬀerent representa- this choice was to have a separate class of sources poten- tive types of galaxies. However, here we use the q100 param- tially hosting nuclear activity for which there is a signiﬁ- eter not to build a clean sample of radio-excess sources, but cant possibility of contamination by non-AGN sources. All rather to mainly pinpoint embedded nuclear activity (RI- but three (77 percent, 10/13) of the RE-AGN-candidates AGN) among the SF/hyb sources, identiﬁed via IR criteria. are conﬁrmed as AGN by the X-ray luminosity or by their Moreover, as we show below, the SF/hyb sources charac- radio-excess. We conclude that the diﬀerence between RE- terised by a radio excess are in a range of redshift and q100 AGN and RE-AGN-candidates is mainly due to the AGN values supporting an AGN contribution to the observed ra- dominance in the IR, but it is reasonable to consider all dio emission regardless of the possible redshift evolution of these sources as true AGN. the q parameter. A full IR SED model ﬁtting applied to 100 a larger sample (selected from 1.4 GHz observations) will be presented in Paper II. 6.7 1.4 GHz VLBI detections Fifty-two sources (out of the 77 radio sources with full IRAC photometry) have a Herschel-PACS detection at The most direct conﬁrmation of a radio AGN is pro- 100 µm, and for the remaining 25 objects we use a 3σ upper vided by the observation of high brightness temperature limit of 1.0 mJy (Magnelli et al. 2013, 2015). The 1.4 GHz components on milli-arcsec (mas) angular scales as probed ﬂux densities are taken from Morrison et al. (2010) for all by Very Long Baseline Interferometry (VLBI). Although in but three sources which are detected only at 5.5 GHz. For the local Universe compact and intense radio emission on these three sources we derive the 1.4 GHz ﬂux density from mas-scales might be also associated with Supernovae and that measured at 5.5 GHz using a spectral index of 0.7. Supernova Remnants, these are diﬃcult to detect with VLBI In Fig. 10 we plot the observed q values against red- at z > 0.1. 100 shift for all the catalogued sources with an infrared clas- Twenty GOODS-N sources were detected by recent siﬁcation. Considering q < 1.5 as a proxy for selecting global VLBI observations at 1.6 GHz with S > 50 µJy 100 V LBI radio-excess sources, we have 31 radio-excess sources (41 and an angular resolution of 4 mas (Radcliﬀe et al. 2016). percent of the sub-sample with full IRAC coverage). As ex- Ten of them were previously detected with VLBI at 1.4 GHz pected, radiatively-ineﬃcient AGN are typically associated by Chi et al. (2013). VLBI cores are found to account for, with radio-excess sources: 87 percent (13/15) of this class of on average, 30 percent of the total 1.6 GHz emission. At sources have q < 1.5 and for the two remaining sources 5.5 GHz, we catalogued 19 of the 20 sources with a VLBI 100 the q value is an upper limit. About one third (11/32) of detection. The remaining source is located outside the re- 100 RE-AGN (including also the RE-AGN-candidates), are asso- gion covered by our 5.5 GHz mosaic. About half (9/19) of ciated with radio-excess sources. This signiﬁcant fraction is the 5.5 GHz sources with VLBI detections belong to the class not entirely surprising as we are dealing with radio-selected of RI-AGN and hence are associated with optically-passive RE-AGN. However, the number of radio-excess RE-AGN galaxies. This means that about 60 percent (9/15) of the could vary signiﬁcantly, since many of those at high redshift radio sources belonging to this class have a VLBI detec- are close to the q threshold adopted here. Moreover, at tion, conﬁrming the presence of radio bright compact cores 100 z >∼ 2 the criterion q < 1.5 could be too simplistic and in RI-AGN. Regarding RE-AGN, about 26 percent (5/19) 100 a redshift evolution of this parameter should be taken into have a VLBI detection, with four of them classiﬁed as radio- account. In any case, this does not inﬂuence the census of excess sources with a redshift z <∼ 1. The remaining object ∼ nuclear activity as most of the sources at such redshifts are is a RE-AGN with q = 1.7 and z 0.6, very close to 100 already classiﬁed as AGN on the basis of the IR colours or the dividing line deﬁning radio-excess sources). Only one of hard-band X-ray emission. We also note that RE-AGN is the RE-AGN-candidates is detected by VLBI, and is also a the only class for which there is a clear link between the radio-excess source and X-ray AGN. Among the sources IR- radio-excess and the X-ray emission: all the radio sources classiﬁed as SF/hyb, three galaxies have a VLBI detection. classiﬁed as RE-AGN and showing a radio-excess (7 out of All three were classiﬁed as radio-excess sources and show a 19) are also X-ray AGN. bright, compact core in the VLA observations. Finally, one MNRAS 000, 000–000 (0000)

text: D.

text: Guidetti

text: et

text: al.

text: ties,

text: from

text: normal,

text: radio-quiet

text: spirals

text: to

text: ultra-luminous

text: IR

text: We

text: ﬁnd

text: ﬁve

text: radio

text: sources

text: at

text: z

text: classiﬁed

text: as

text: galaxies

text: (ULIRGs)

text: (Yun

text: et

text: al.

text: Garrett

text: Appleton

text: SF/hyb

text: systems

text: and

text: not

text: strong

text: X-ray

text: emitters,

text: that

text: show

text: a

text: et

text: al.

text: and

text: is

text: one

text: of

text: the

text: most

text: useful

text: diagnostic

text: tools

text: clear

text: radio-excess

text: (q

text: for

text: four

text: of

text: them).

text: Even

text: taking

text: in

text: revealing

text: excess

text: radio

text: emission

text: exceeding

text: that

text: expected

text: into

text: account

text: the

text: evolution

text: with

text: redshift

text: of

text: q

text: all

text: these

text: from

text: pure

text: star-formation

text: processes.

text: sources

text: would

text: fall

text: in

text: the

text: radio-excess

text: region

text: (e.g.

text: see

text: in

text: Several

text: studies

text: have

text: replaced

text: the

text: FIR

text: ﬂux

text: with

text: the

text: Del

text: Moro

text: et

text: al.

text: Finally,

text: there

text: is

text: an

text: IR-excess

text: source

text: monochromatic

text: ﬂux

text: at

text: µm

text: (e.g.

text: Bonzini

text: et

text: al.

text: or

text: at

text: at

text: z

text: here

text: the

text: IR

text: emission

text: should

text: be

text: dominated

text: by

text: µm

text: (e.g.

text: Del

text: Moro

text: et

text: al.

text: as

text: a

text: proxy

text: for

text: the

text: FIR

text: an

text: AGN

text: and

text: indeed

text: the

text: sources

text: is

text: classiﬁed

text: as

text: such

text: on

text: the

text: emission.

text: In

text: particular,

text: Del

text: Moro

text: et

text: al.

text: showed

text: that

text: a

text: basis

text: of

text: both

text: the

text: IR

text: and

text: X-ray

text: diagnostics.

text: It

text: does

text: not

text: show

text: simple

text: cut

text: at

text: q

text: (where

text: q

text: log(S

text: /S

text: a

text: radio

text: excess

text: as

text: typically

text: observed

text: in

text: many

text: RE-AGN.

text: The

text: selects

text: percent

text: of

text: the

text: radio-excess

text: sources

text: deﬁned

text: us-

text: radio

text: excess

text: sources

text: are

text: shown

text: as

text: black

text: squares

text: in

text: Fig.

text: ing

text: the

text: total

text: FIR.

text: Following

text: Del

text: Moro

text: et

text: al.

text: we

text: use

text: In

text: Table

text: we

text: list,

text: for

text: each

text: of

text: the

text: IR

text: classes,

text: the

text: number

text: to

text: identify

text: radio-excess

text: sources

text: in

text: our

text: radio

text: se-

text: of

text: X-ray

text: AGN,

text: the

text: number

text: of

text: radio-excess

text: sources

text: and

text: the

text: lected

text: sample.

text: Clearly,

text: such

text: a

text: criterion

text: is

text: simplistic

text: and

text: does

text: number

text: of

text: sources

text: which

text: are

text: both

text: X-ray

text: and

text: radio-excess

text: not

text: take

text: into

text: account

text: the

text: z

text: dependence,

text: such

text: that

text: it

text: may

text: suf-

text: AGN.

text: At

text: this

text: point

text: it

text: is

text: important

text: to

text: recall

text: that

text: we

text: have

text: fer

text: of

text: contamination,

text: especially

text: at

text: high

text: redshift,

text: by

text: strong

text: IR

text: classiﬁed

text: as

text: RE-AGN-candidates

text: those

text: sources

text: selected

text: as

text: sources

text: (i.e.

text: hyper-luminous

text: IR

text: galaxies):

text: ideally,

text: should

text: AGN

text: by

text: only

text: one

text: of

text: the

text: four

text: IR

text: diagnostics.

text: The

text: reason

text: for

text: be

text: compared

text: to

text: the

text: expected

text: tracks

text: for

text: diﬀerent

text: representa-

text: this

text: choice

text: was

text: to

text: have

text: a

text: separate

text: class

text: of

text: sources

text: poten-

text: tive

text: types

text: of

text: galaxies.

text: However,

text: here

text: we

text: use

text: the

text: param-

text: tially

text: hosting

text: nuclear

text: activity

text: for

text: which

text: there

text: is

text: a

text: signiﬁ-

text: eter

text: not

text: to

text: build

text: a

text: clean

text: sample

text: of

text: radio-excess

text: sources,

text: but

text: cant

text: possibility

text: of

text: contamination

text: by

text: non-AGN

text: sources.

text: All

text: rather

text: to

text: mainly

text: pinpoint

text: embedded

text: nuclear

text: activity

text: (RI-

text: but

text: three

text: percent,

text: of

text: the

text: RE-AGN-candidates

text: AGN)

text: among

text: the

text: SF/hyb

text: sources,

text: identiﬁed

text: via

text: IR

text: criteria.

text: are

text: conﬁrmed

text: as

text: AGN

text: by

text: the

text: X-ray

text: luminosity

text: or

text: by

text: their

text: Moreover,

text: as

text: we

text: show

text: below,

text: the

text: SF/hyb

text: sources

text: charac-

text: radio-excess.

text: We

text: conclude

text: that

text: the

text: diﬀerence

text: between

text: RE-

text: terised

text: by

text: a

text: radio

text: excess

text: are

text: in

text: a

text: range

text: of

text: redshift

text: and

text: AGN

text: and

text: RE-AGN-candidates

text: is

text: mainly

text: due

text: to

text: the

text: AGN

text: values

text: supporting

text: an

text: AGN

text: contribution

text: to

text: the

text: observed

text: ra-

text: dominance

text: in

text: the

text: IR,

text: but

text: it

text: is

text: reasonable

text: to

text: consider

text: all

text: dio

text: emission

text: regardless

text: of

text: the

text: possible

text: redshift

text: evolution

text: of

text: these

text: sources

text: as

text: true

text: AGN.

text: the

text: q

text: parameter.

text: A

text: full

text: IR

text: SED

text: model

text: ﬁtting

text: applied

text: to

text: a

text: larger

text: sample

text: (selected

text: from

text: GHz

text: observations)

text: will

text: be

text: presented

text: in

text: Paper

text: II.

text: GHz

text: VLBI

text: detections

text: Fifty-two

text: sources

text: (out

text: of

text: the

text: radio

text: sources

text: with

text: full

text: IRAC

text: photometry)

text: have

text: a

text: Herschel-PACS

text: detection

text: at

text: The

text: most

text: direct

text: conﬁrmation

text: of

text: a

text: radio

text: AGN

text: is

text: pro-

text: µm,

text: and

text: for

text: the

text: remaining

text: objects

text: we

text: use

text: a

text: 3σ

text: upper

text: vided

text: by

text: the

text: observation

text: of

text: high

text: brightness

text: temperature

text: limit

text: of

text: mJy

text: (Magnelli

text: et

text: al.

text: The

text: GHz

text: components

text: on

text: milli-arcsec

text: (mas)

text: angular

text: scales

text: as

text: probed

text: ﬂux

text: densities

text: are

text: taken

text: from

text: Morrison

text: et

text: al.

text: for

text: all

text: by

text: Very

text: Long

text: Baseline

text: Interferometry

text: (VLBI).

text: Although

text: in

text: but

text: three

text: sources

text: which

text: are

text: detected

text: only

text: at

text: GHz.

text: For

text: the

text: local

text: Universe

text: compact

text: and

text: intense

text: radio

text: emission

text: on

text: these

text: three

text: sources

text: we

text: derive

text: the

text: GHz

text: ﬂux

text: density

text: from

text: mas-scales

text: might

text: be

text: also

text: associated

text: with

text: Supernovae

text: and

text: that

text: measured

text: at

text: GHz

text: using

text: a

text: spectral

text: index

text: of

text: Supernova

text: Remnants,

text: these

text: are

text: diﬃcult

text: to

text: detect

text: with

text: VLBI

text: In

text: Fig.

text: we

text: plot

text: the

text: observed

text: q

text: values

text: against

text: red-

text: at

text: z

text: shift

text: for

text: all

text: the

text: catalogued

text: sources

text: with

text: an

text: infrared

text: clas-

text: Twenty

text: GOODS-N

text: sources

text: were

text: detected

text: by

text: recent

text: siﬁcation.

text: Considering

text: q

text: as

text: a

text: proxy

text: for

text: selecting

text: global

text: VLBI

text: observations

text: at

text: GHz

text: with

text: S

text: µJy

text: V

text: LBI

text: radio-excess

text: sources,

text: we

text: have

text: radio-excess

text: sources

text: and

text: an

text: angular

text: resolution

text: of

text: mas

text: (Radcliﬀe

text: et

text: al.

text: percent

text: of

text: the

text: sub-sample

text: with

text: full

text: IRAC

text: coverage).

text: As

text: ex-

text: Ten

text: of

text: them

text: were

text: previously

text: detected

text: with

text: VLBI

text: at

text: GHz

text: pected,

text: radiatively-ineﬃcient

text: AGN

text: are

text: typically

text: associated

text: by

text: Chi

text: et

text: al.

text: VLBI

text: cores

text: are

text: found

text: to

text: account

text: for,

text: with

text: radio-excess

text: sources:

text: percent

text: of

text: this

text: class

text: of

text: on

text: average,

text: percent

text: of

text: the

text: total

text: GHz

text: emission.

text: At

text: sources

text: have

text: q

text: and

text: for

text: the

text: two

text: remaining

text: sources

text: GHz,

text: we

text: catalogued

text: of

text: the

text: sources

text: with

text: a

text: VLBI

text: the

text: q

text: value

text: is

text: an

text: upper

text: limit.

text: About

text: one

text: third

text: of

text: detection.

text: The

text: remaining

text: source

text: is

text: located

text: outside

text: the

text: re-

text: RE-AGN

text: (including

text: also

text: the

text: RE-AGN-candidates),

text: are

text: asso-

text: gion

text: covered

text: by

text: our

text: GHz

text: mosaic.

text: About

text: half

text: of

text: ciated

text: with

text: radio-excess

text: sources.

text: This

text: signiﬁcant

text: fraction

text: is

text: the

text: GHz

text: sources

text: with

text: VLBI

text: detections

text: belong

text: to

text: the

text: class

text: not

text: entirely

text: surprising

text: as

text: we

text: are

text: dealing

text: with

text: radio-selected

text: of

text: RI-AGN

text: and

text: hence

text: are

text: associated

text: with

text: optically-passive

text: RE-AGN.

text: However,

text: the

text: number

text: of

text: radio-excess

text: RE-AGN

text: galaxies.

text: This

text: means

text: that

text: about

text: percent

text: of

text: the

text: could

text: vary

text: signiﬁcantly,

text: since

text: many

text: of

text: those

text: at

text: high

text: redshift

text: radio

text: sources

text: belonging

text: to

text: this

text: class

text: have

text: a

text: VLBI

text: detec-

text: are

text: close

text: to

text: the

text: q

text: threshold

text: adopted

text: here.

text: Moreover,

text: at

text: tion,

text: conﬁrming

text: the

text: presence

text: of

text: radio

text: bright

text: compact

text: cores

text: z

text: the

text: criterion

text: q

text: could

text: be

text: too

text: simplistic

text: and

text: in

text: RI-AGN.

text: Regarding

text: RE-AGN,

text: about

text: percent

text: a

text: redshift

text: evolution

text: of

text: this

text: parameter

text: should

text: be

text: taken

text: into

text: have

text: a

text: VLBI

text: detection,

text: with

text: four

text: of

text: them

text: classiﬁed

text: as

text: radio-

text: account.

text: In

text: any

text: case,

text: this

text: does

text: not

text: inﬂuence

text: the

text: census

text: of

text: excess

text: sources

text: with

text: a

text: redshift

text: z

text: The

text: remaining

text: object

text: nuclear

text: activity

text: as

text: most

text: of

text: the

text: sources

text: at

text: such

text: redshifts

text: are

text: is

text: a

text: RE-AGN

text: with

text: q

text: and

text: z

text: very

text: close

text: to

text: already

text: classiﬁed

text: as

text: AGN

text: on

text: the

text: basis

text: of

text: the

text: IR

text: colours

text: or

text: the

text: dividing

text: line

text: deﬁning

text: radio-excess

text: sources).

text: Only

text: one

text: of

text: hard-band

text: X-ray

text: emission.

text: We

text: also

text: note

text: that

text: RE-AGN

text: is

text: the

text: RE-AGN-candidates

text: is

text: detected

text: by

text: VLBI,

text: and

text: is

text: also

text: a

text: the

text: only

text: class

text: for

text: which

text: there

text: is

text: a

text: clear

text: link

text: between

text: the

text: radio-excess

text: source

text: and

text: X-ray

text: AGN.

text: Among

text: the

text: sources

text: IR-

text: radio-excess

text: and

text: the

text: X-ray

text: emission:

text: all

text: the

text: radio

text: sources

text: classiﬁed

text: as

text: SF/hyb,

text: three

text: galaxies

text: have

text: a

text: VLBI

text: detection.

text: classiﬁed

text: as

text: RE-AGN

text: and

text: showing

text: a

text: radio-excess

text: out

text: of

text: All

text: three

text: were

text: classiﬁed

text: as

text: radio-excess

text: sources

text: and

text: show

text: a

text: are

text: also

text: X-ray

text: AGN.

text: bright,

text: compact

text: core

text: in

text: the

text: VLA

text: observations.

text: Finally,

text: one

text: MNRAS


text: VLA 5.5 GHz observations of the GOODS-North Field 17 in the present paper that allowed to detect lower luminosity DG, MB, IP acknowledge support from PRIN-INAF AGN with respect to the ancillary data used in M05. 2014 (PI M. Bondi). DG and IP acknowledge support of the 8 CONCLUSIONS Ministry of Foreign Aﬀairs and International Cooperation, Directorate General for the Country Promotion (Bilateral Using ultra-deep sub-arcsec-resolution radio observa- Grant Agreement ZA14GR02 - Mapping the Universe on tions at 5.5 GHz obtained with the VLA in the framework of the Pathway to SKA) the eMERGE legacy project, we have produced a mosaic (in- REFERENCES cluding seven diﬀerent pointings) with a median r.m.s. noise − × of 3 µJy beam 1 and an angular resolution of 0.56 0.47 Alexander, D. M., Bauer, F. E., Brandt, W. N., et al. 2003, AJ, arcsec2 over a circular region with a diameter of 14 arcmin 126, 539 centred on the GOODS-N radio ﬁeld. Appleton, P. N., Fadda, D. T., Marleau, F. R., et al. 2004, ApJS, 154, 147 The main results presented in this paper can can be Ashby, M. L. N., Willner, S. P., Fazio, G. G., et al. 2015, ApJS, summarised as follows: • 218, 33 We extracted a catalogue containing 94 radio sources Barger, A. J., Cowie, L. L., & Wang, W.-H. 2008, ApJ, 689, 687- above the local 5σ threshold, with about 50 percent of the − − 708 sources in the range 10 30 µJy beam− 1 and less than 20 Bauer, F. E., Alexander, D. M., Brandt, W. N., et al. 2002, AJ, percent with peak ﬂux > 100 µJy beam 1. About 60 percent 124, 2351 (56/94) of the radio sources are classiﬁed as resolved on the Bauer, F. E., Alexander, D. M., Brandt, W. N., et al. 2004, AJ, basis of the total-to-peak ﬂux ratio versus SNR plot. 128, 2048 • We used deep NIR catalogues, mainly Wang et al. Best, P. N., Kaiser, C. R., Heckman, T. M., & Kauﬀmann, G. 20 (2010), but also Kajisawa et al. (2011) and Skelton et al. Bondi, M., Ciliegi, P., Zamorani, G., et al. 2003, A&A, 403, 857 Bondi, M., Ciliegi, P., Venturi, T., et al. 2007, A&A, 463, 519 (2014), to identify the radio sources. We ﬁnd that 88 per- Bondi, M., Ciliegi, P., Schinnerer, E., et al. 2008, ApJ, 681, 1129- cent (83/94) of the radio catalogue have secure NIR iden- 1135 tiﬁcations, with the fraction raising to 96 percent (76/79) Bonzini, M., Padovani, P., Mainieri, V., et al. 2013, MNRAS, 436, when only the radio sources above 5.5σ are considered. • 3759 Redshift information is available for 95 percent (79/83) Brammer, G. B., van Dokkum, P. G., & Coppi, P. 2008, ApJ, 686, of the NIR identiﬁed radio sources (51 redshifts are spectro- 1503-1513 scopic and 28 photometric). The median redshift is zmed = Cardamone, C. N., Urry, C. M., Damen, M., et al. 2008, ApJ, 1.32. 680, 130-142 • We used multi-band AGN diagnostics (IR colour-colour Casey, C. M. 2012, MNRAS, 425, 3094 plots, X-ray luminosity, radio-excess parameter and VLBI Chi, S., Barthel, P. D., & Garrett, M. A. 2013, A&A, 550, A68 detection) to separate AGN-driven (both radiatively eﬃ- Condon, J. J. 1997, PASP, 109, 166 Condon, J. J., Cotton, W. D., Fomalont, E. B., et al. 2012, ApJ, cient and ineﬃcient) radio sources from SFGs in a subsam- 758, 23 ple of 77 radio sources with a detection in all the four IRAC Condon, J. J., 2015, ArXiv e-prints, arXiv:1502.05616 bands. We ﬁnd that 79 percent (61/77) of the sources show Cowie, L. L., Barger, A. J., Bautz, M. W., et al. 2001, ApJ, 551, evidence for nuclear activity and this fraction is about 92 L9 percent if we consider only the sources with redshift z > 1.5. Croton, D. J., Springel, V., White, S. D. M., et al. 2006, MNRAS, Such a large fraction of AGN is unusual considering we are 365, 11 sampling a population of radio sources with a median peak Delvecchio, I., Smolˇci´c, V., Zamorani, G., et al. 2017, ArXiv e- (cid:39) − brightness of 20 µJy beam 1. prints • Our conclusion is that we are missing SFGs because Del Moro, A., Alexander, D. M., Mullaney, J. R., et al. 2013, of the limited surface brightness sensitivity, due to the lim- A&A, 549, A59 Di Matteo, T., Springel, V., & Hernquist, L. 2005, Nature, 433, ited availability of short spacings. This favours the detec- 604 tion of compact kpc/sub-kpc radio sources at the expense Donley, J. L., Koekemoer, A. M., Brusa, M., et al. 2012, ApJ, of sources with radio emission distributed on scales of several 748, 142 kpc. Indeed, the AGN populations (both RE- and RI-AGN) (cid:39) − Elbaz, D., Dickinson, M., Hwang, H. S., et al. 2011, A&A, 533, have very similar median angular sizes ( (cid:39)0.2 0.3 arcsec A119 for z < 1.5), the SFGs have larger sizes ( 0.8 arcsec for Fabian, A. C. 2012, ARA&A, 50, 455 z < 1.5). Finally, AGN-hosting radio sources (RE- and RI- Feigelson, E. D., & Nelson, P. I. 1985, ApJ, 293, 192 AGN) dominate the population of our catalogue at all ﬂux Ferrarese, L. 2002, ApJ, 578, 90 density levels. Garrett, M. A. 2002, A&A, 384, L19 Gebhardt, K., Bender, R., Bower, G., et al. 2000, ApJ, 539, L13 The afore-mentioned selection eﬀects need to be taken Giavalisco, M., Ferguson, H. C., Koekemoer, A. M., et al. 2004, into account in planning future surveys. Such eﬀects will be ApJ, 600, L93 further discussed in a forthcoming paper based on a com- Griﬃn, M. J., Abergel, A., Abreu, A., et al. 2010, A&A, 518, L3 parative analysis of radio-selected samples with diﬀerent an- Gruppioni, C., Mignoli, M., & Zamorani, G. 1999, MNRAS, 304, 199 gular resolutions and frequency, but comparable depths to Guidetti, D., Laing, R. A., Bridle, A. H., Parma, P., & Gregorini, GOODS-N. In that paper we will also discuss the origin of L. 2011, MNRAS, 413, 2525 the radio emission in RE-AGN. Guidetti, D., Laing, R. A., Croston, J. H., Bridle, A. H., & Parma, P. 2012, MNRAS, 423, 1335 Guidetti, D., Bondi, M., Prandoni, I., Beswick, R. J., Muxlow, T. W. B., Wrigley, N., Smail, I., McHardy, I. 2013, MNRAS, ACKNOWLEDGEMENTS 432, 2798 MNRAS 000, 000–000 (0000)

text: VLA

text: GHz

text: observations

text: of

text: the

text: GOODS-North

text: Field

text: in

text: the

text: present

text: paper

text: that

text: allowed

text: to

text: detect

text: lower

text: luminosity

text: DG,

text: MB,

text: IP

text: acknowledge

text: support

text: from

text: PRIN-INAF

text: AGN

text: with

text: respect

text: to

text: the

text: ancillary

text: data

text: used

text: in

text: (PI

text: M.

text: Bondi).

text: DG

text: and

text: IP

text: acknowledge

text: support

text: of

text: the

text: CONCLUSIONS

text: Ministry

text: of

text: Foreign

text: Aﬀairs

text: and

text: International

text: Cooperation,

text: Directorate

text: General

text: for

text: the

text: Country

text: Promotion

text: (Bilateral

text: Using

text: ultra-deep

text: sub-arcsec-resolution

text: radio

text: observa-

text: Grant

text: Agreement

text: Mapping

text: the

text: Universe

text: on

text: tions

text: at

text: GHz

text: obtained

text: with

text: the

text: VLA

text: in

text: the

text: framework

text: of

text: the

text: Pathway

text: to

text: SKA)

text: the

text: eMERGE

text: legacy

text: project,

text: we

text: have

text: produced

text: a

text: mosaic

text: (in-

text: REFERENCES

text: cluding

text: seven

text: diﬀerent

text: pointings)

text: with

text: a

text: median

text: r.m.s.

text: noise

text: of

text: µJy

text: beam

text: and

text: an

text: angular

text: resolution

text: of

text: Alexander,

text: D.

text: M.,

text: Bauer,

text: F.

text: E.,

text: Brandt,

text: W.

text: N.,

text: et

text: al.

text: AJ,

text: over

text: a

text: circular

text: region

text: with

text: a

text: diameter

text: of

text: arcmin

text: centred

text: on

text: the

text: GOODS-N

text: radio

text: ﬁeld.

text: Appleton,

text: P.

text: N.,

text: Fadda,

text: D.

text: T.,

text: Marleau,

text: F.

text: R.,

text: et

text: al.

text: ApJS,

text: The

text: main

text: results

text: presented

text: in

text: this

text: paper

text: can

text: can

text: be

text: Ashby,

text: M.

text: L.

text: N.,

text: Willner,

text: S.

text: P.,

text: Fazio,

text: G.

text: G.,

text: et

text: al.

text: ApJS,

text: summarised

text: as

text: follows:

text: We

text: extracted

text: a

text: catalogue

text: containing

text: radio

text: sources

text: Barger,

text: A.

text: J.,

text: Cowie,

text: L.

text: L.,

text: Wang,

text: W.-H.

text: ApJ,

text: above

text: the

text: local

text: 5σ

text: threshold,

text: with

text: about

text: percent

text: of

text: the

text: sources

text: in

text: the

text: range

text: µJy

text: beam−

text: and

text: less

text: than

text: Bauer,

text: F.

text: E.,

text: Alexander,

text: D.

text: M.,

text: Brandt,

text: W.

text: N.,

text: et

text: al.

text: AJ,

text: percent

text: with

text: peak

text: ﬂux

text: µJy

text: beam

text: About

text: percent

text: of

text: the

text: radio

text: sources

text: are

text: classiﬁed

text: as

text: resolved

text: on

text: the

text: Bauer,

text: F.

text: E.,

text: Alexander,

text: D.

text: M.,

text: Brandt,

text: W.

text: N.,

text: et

text: al.

text: AJ,

text: basis

text: of

text: the

text: total-to-peak

text: ﬂux

text: ratio

text: versus

text: SNR

text: plot.

text: We

text: used

text: deep

text: NIR

text: catalogues,

text: mainly

text: Wang

text: et

text: al.

text: Best,

text: P.

text: N.,

text: Kaiser,

text: C.

text: R.,

text: Heckman,

text: T.

text: M.,

text: Kauﬀmann,

text: G.

text: but

text: also

text: Kajisawa

text: et

text: al.

text: and

text: Skelton

text: et

text: al.

text: Bondi,

text: M.,

text: Ciliegi,

text: P.,

text: Zamorani,

text: G.,

text: et

text: al.

text: A&A,

text: Bondi,

text: M.,

text: Ciliegi,

text: P.,

text: Venturi,

text: T.,

text: et

text: al.

text: A&A,

text: to

text: identify

text: the

text: radio

text: sources.

text: We

text: ﬁnd

text: that

text: per-

text: Bondi,

text: M.,

text: Ciliegi,

text: P.,

text: Schinnerer,

text: E.,

text: et

text: al.

text: ApJ,

text: cent

text: of

text: the

text: radio

text: catalogue

text: have

text: secure

text: NIR

text: iden-

text: tiﬁcations,

text: with

text: the

text: fraction

text: raising

text: to

text: percent

text: Bonzini,

text: M.,

text: Padovani,

text: P.,

text: Mainieri,

text: V.,

text: et

text: al.

text: MNRAS,

text: when

text: only

text: the

text: radio

text: sources

text: above

text: are

text: considered.

text: Redshift

text: information

text: is

text: available

text: for

text: percent

text: Brammer,

text: G.

text: B.,

text: van

text: Dokkum,

text: P.

text: G.,

text: Coppi,

text: P.

text: ApJ,

text: of

text: the

text: NIR

text: identiﬁed

text: radio

text: sources

text: redshifts

text: are

text: spectro-

text: scopic

text: and

text: photometric).

text: The

text: median

text: redshift

text: is

text: zmed

text: Cardamone,

text: C.

text: N.,

text: Urry,

text: C.

text: M.,

text: Damen,

text: M.,

text: et

text: al.

text: ApJ,

text: We

text: used

text: multi-band

text: AGN

text: diagnostics

text: (IR

text: colour-colour

text: Casey,

text: C.

text: M.

text: MNRAS,

text: plots,

text: X-ray

text: luminosity,

text: radio-excess

text: parameter

text: and

text: VLBI

text: Chi,

text: S.,

text: Barthel,

text: P.

text: D.,

text: Garrett,

text: M.

text: A.

text: A&A,

text: detection)

text: to

text: separate

text: AGN-driven

text: (both

text: radiatively

text: eﬃ-

text: Condon,

text: J.

text: J.

text: PASP,

text: Condon,

text: J.

text: J.,

text: Cotton,

text: W.

text: D.,

text: Fomalont,

text: E.

text: B.,

text: et

text: al.

text: ApJ,

text: cient

text: and

text: ineﬃcient)

text: radio

text: sources

text: from

text: SFGs

text: in

text: a

text: subsam-

text: ple

text: of

text: radio

text: sources

text: with

text: a

text: detection

text: in

text: all

text: the

text: four

text: IRAC

text: Condon,

text: J.

text: J.,

text: ArXiv

text: e-prints,

text: bands.

text: We

text: ﬁnd

text: that

text: percent

text: of

text: the

text: sources

text: show

text: Cowie,

text: L.

text: L.,

text: Barger,

text: A.

text: J.,

text: Bautz,

text: M.

text: W.,

text: et

text: al.

text: ApJ,

text: evidence

text: for

text: nuclear

text: activity

text: and

text: this

text: fraction

text: is

text: about

text: percent

text: if

text: we

text: consider

text: only

text: the

text: sources

text: with

text: redshift

text: z

text: Croton,

text: D.

text: J.,

text: Springel,

text: V.,

text: White,

text: S.

text: D.

text: M.,

text: et

text: al.

text: MNRAS,

text: Such

text: a

text: large

text: fraction

text: of

text: AGN

text: is

text: unusual

text: considering

text: we

text: are

text: sampling

text: a

text: population

text: of

text: radio

text: sources

text: with

text: a

text: median

text: peak

text: Delvecchio,

text: I.,

text: Smolˇci´c,

text: V.,

text: Zamorani,

text: G.,

text: et

text: al.

text: ArXiv

text: e-

text: brightness

text: of

text: µJy

text: beam

text: prints

text: Our

text: conclusion

text: is

text: that

text: we

text: are

text: missing

text: SFGs

text: because

text: Del

text: Moro,

text: A.,

text: Alexander,

text: D.

text: M.,

text: Mullaney,

text: J.

text: R.,

text: et

text: al.

text: of

text: the

text: limited

text: surface

text: brightness

text: sensitivity,

text: due

text: to

text: the

text: lim-

text: A&A,

text: Di

text: Matteo,

text: T.,

text: Springel,

text: V.,

text: Hernquist,

text: L.

text: Nature,

text: ited

text: availability

text: of

text: short

text: spacings.

text: This

text: favours

text: the

text: detec-

text: tion

text: of

text: compact

text: kpc/sub-kpc

text: radio

text: sources

text: at

text: the

text: expense

text: Donley,

text: J.

text: L.,

text: Koekemoer,

text: A.

text: M.,

text: Brusa,

text: M.,

text: et

text: al.

text: ApJ,

text: of

text: sources

text: with

text: radio

text: emission

text: distributed

text: on

text: scales

text: of

text: several

text: kpc.

text: Indeed,

text: the

text: AGN

text: populations

text: (both

text: RE-

text: and

text: RI-AGN)

text: Elbaz,

text: D.,

text: Dickinson,

text: M.,

text: Hwang,

text: H.

text: S.,

text: et

text: al.

text: A&A,

text: have

text: very

text: similar

text: median

text: angular

text: sizes

text: arcsec

text: for

text: z

text: the

text: SFGs

text: have

text: larger

text: sizes

text: arcsec

text: for

text: Fabian,

text: A.

text: C.

text: ARA&A,

text: z

text: Finally,

text: AGN-hosting

text: radio

text: sources

text: (RE-

text: and

text: RI-

text: Feigelson,

text: E.

text: D.,

text: Nelson,

text: P.

text: I.

text: ApJ,

text: AGN)

text: dominate

text: the

text: population

text: of

text: our

text: catalogue

text: at

text: all

text: ﬂux

text: Ferrarese,

text: L.

text: ApJ,

text: density

text: levels.

text: Garrett,

text: M.

text: A.

text: A&A,

text: Gebhardt,

text: K.,

text: Bender,

text: R.,

text: Bower,

text: G.,

text: et

text: al.

text: ApJ,

text: The

text: afore-mentioned

text: selection

text: eﬀects

text: need

text: to

text: be

text: taken

text: Giavalisco,

text: M.,

text: Ferguson,

text: H.

text: C.,

text: Koekemoer,

text: A.

text: M.,

text: et

text: al.

text: into

text: account

text: in

text: planning

text: future

text: surveys.

text: Such

text: eﬀects

text: will

text: be

text: ApJ,

text: further

text: discussed

text: in

text: a

text: forthcoming

text: paper

text: based

text: on

text: a

text: com-

text: Griﬃn,

text: M.

text: J.,

text: Abergel,

text: A.,

text: Abreu,

text: A.,

text: et

text: al.

text: A&A,

text: parative

text: analysis

text: of

text: radio-selected

text: samples

text: with

text: diﬀerent

text: an-

text: Gruppioni,

text: C.,

text: Mignoli,

text: M.,

text: Zamorani,

text: G.

text: MNRAS,

text: gular

text: resolutions

text: and

text: frequency,

text: but

text: comparable

text: depths

text: to

text: Guidetti,

text: D.,

text: Laing,

text: R.

text: A.,

text: Bridle,

text: A.

text: H.,

text: Parma,

text: P.,

text: Gregorini,

text: GOODS-N.

text: In

text: that

text: paper

text: we

text: will

text: also

text: discuss

text: the

text: origin

text: of

text: L.

text: MNRAS,

text: the

text: radio

text: emission

text: in

text: RE-AGN.

text: Guidetti,

text: D.,

text: Laing,

text: R.

text: A.,

text: Croston,

text: J.

text: H.,

text: Bridle,

text: A.

text: H.,

text: Parma,

text: P.

text: MNRAS,

text: Guidetti,

text: D.,

text: Bondi,

text: M.,

text: Prandoni,

text: I.,

text: Beswick,

text: R.

text: J.,

text: Muxlow,

text: T.

text: W.

text: B.,

text: Wrigley,

text: N.,

text: Smail,

text: I.,

text: McHardy,

text: I.

text: MNRAS,

text: ACKNOWLEDGEMENTS

text: MNRAS