text: 2 and Earth-term events. In Sections 7 and 8, we present sion alone become more signiﬁcant (Cordes & Shannon the results of our pulsar-term and Earth-term searches, 2010). Furthermore, this pulsar is directly behind a com- respectively. In Section 9, we place upper bounds on plex region of H-II associated with ζ-Ophiuchi, a mas- BWM rates and amplitudes. In Section 10 we summa- sive, runaway O-type star spinning very near breakup rize our key results and oﬀer concluding remarks. We (Gaustad et al. 2001; Villamariz & Herrero 2005). This provide a summary of our important notational elements intervening H-II region could conceivably contribute to in a table of key symbols in an appendix. non-trivial and currently unaccounted-for chromatic ef- − 2. PULSAR TIMING DATA SET fects on the timing behavior of J1643 1224. In this section, we discuss several aspects of the data 3. SIGNAL MODEL that are relevant to our analysis. For a more thorough For a given pulsar, the timing perturbation from a description of the data, see Demorest (2007) and Demor- BWM of amplitude h is well-modeled as B est et al. (2013). The ﬁve-year data set consists of pulse times-of-arrival − − − ∆t(t) = h B(θ, φ) [(t t )Θ(t t ) (TOAs) collected at approximately monthly intervals for B − 0 − 0 each of 17 millisecond pulsars (MSPs). All observations (t t )Θ(t t )] . (1) 1 1 were done with the Arecibo radio telescope and the Green Bank Telescope (GBT). Of these 17 MSPs, those visi- − The function B(θ, φ) = (1/2)(1 cos θ) cos (2φ) ranges ble to Arecibo were observed with Arecibo; all others − between 1 and 1 and is common to all pulsar timing were observed with the GBT. One pulsar, J1713+0747, eﬀorts to detect point-like sources of GWs (Estabrook was observed with both telescopes. All observations were & Wahlquist 1975; Hellings & Downs 1983; Lee et al. done using one of two identical backend systems: the As- 2011). The angle between the direction the burst prop- tronomical Signal Processor (ASP) and the Green Bank agates and the line of sight from Earth to the pulsar is Astronomical Signal Processor (GASP). These backends θ; φ is the angle between the principal polarization vec- performed real-time coherent dedispersion over bands up tor of the wave and the projection of the line of sight to 64 MHz wide and recorded the results averaged over from the Earth to the pulsar onto the plane normal to channels of width 4 MHz each. the wave propagation direction. The BWM encounters For each observing epoch, several TOAs are reported the Earth at a time t and is observed from Earth to from various frequency channels of the 64 MHz band. 0 encounter the pulsar at a time t = t + (l/c)(1 + cos θ) At Arecibo, observations were typically conducted at 1 0 where l is the distance from the Earth to the pulsar (van two widely-separated frequencies (usually 430 MHz and Haasteren & Levin 2010; Cordes & Jenet 2012; Madison 1400 MHz) within one day; at the GBT, observations at et al. 2014). The function Θ is the Heaviside step func- two frequencies (usually 820 MHz and 1400 MHz) were tion. The amplitude of a BWM coming from a SMBHB conducted within several days of each other. Approx- ≡ merger of reduced mass µ M M /(M +M ) ( M and imately contemporaneous observations at multiple fre- 1 2 1 2 1 M are the masses of the black holes in the binary) with quencies allow epoch-to-epoch timing ﬂuctuations caused 2 I a typical inclination angle of = π/3 at a luminosity by changes in dispersion measure (DM) to be accounted (cid:18) (cid:19) (cid:18) (cid:19) distance D from Earth is (Madison et al. 2014) for in the timing model ﬁt (Lam et al. 2014). L Three of the pulsars comprising the ﬁve-year NANOGrav data set, J1853+1308, J1910+1256, and B1953+29, do ≈ × − µ 1 Mpc not have suﬃcient dual-frequency coverage to correct hB 1.5 10 12 109 M(cid:12) D . (2) for timing errors from variations in DM over time; we L exclude these pulsars from our analysis for this reason. − The distances to the pulsars in the NANOGrav array We also exclude the data set for J1600 3053 from our are on the order of kiloparsecs. Our timing baseline is analysis because it is comparatively very short, spanning approximately ﬁve years. Unless θ diﬀers from π by less just two years. Searching for BWMs in such short data ∼ than 3 degrees for a particular pulsar, we expect that sets is feasible in principle, but we avoid it for two rea- t and t will not both fall within our observing window. sons. First, the minimum detectable BWM amplitude 0 1 − Since we consider only 12 pulsars in our analysis, less in a particular data set scales approximately as T 3/2 than 1% of the sky is within 3 degrees of one of our pul- where T is the span of the data set (van Haasteren & sars. Assuming BWMs occur isotropically, there is a less Levin 2010), so we do not anticipate that this data set than 1% chance of both t and t occurring within our will greatly improve our sensitivity to BWMs. Second, 0 1 ﬁve-year observing span if a BWM occurs at all. And in such short data sets, many timing model parameters if a BWM occurred with a small enough angular sepa- are highly covariant with each other (e.g., spin and as- ration from a pulsar in our array that we could see the trometric parameters; Madison et al. 2013) and with any timing perturbation both turn on at t and turn oﬀ at t , BWM signal present. 0 1 − it would only be observed to turn oﬀ in that one pulsar. Finally we exclude the data for J1643 1224 from our So, in each of our pulsar timing data sets, we need only analysis because we believe it contains chromatic tim- to look for evidence of timing perturbations of the form ing biases from unaccounted-for phenomenology in the interstellar medium (ISM; see Figure 2). The DM of − − − − J1643 1224 is approximately 62 pc cm 3, nearly a factor ∆t(t) = h (t t )Θ(t t ), (3) p B B of two greater than any other pulsar in our sample. With high DM pulsars, chromatic timing errors that deviate ± from the ν−2 scaling expected from cold plasma disper- where hp = hBB(θ, φ), what we call the projected burst amplitude, and t is either t or t , what we call B 0 1 ##LTLine## ##LTLine##

text: and

text: Earth-term

text: events.

text: In

text: Sections

text: and

text: we

text: present

text: sion

text: alone

text: become

text: more

text: signiﬁcant

text: (Cordes

text: Shannon

text: the

text: results

text: of

text: our

text: pulsar-term

text: and

text: Earth-term

text: searches,

text: Furthermore,

text: this

text: pulsar

text: is

text: directly

text: behind

text: a

text: com-

text: respectively.

text: In

text: Section

text: we

text: place

text: upper

text: bounds

text: on

text: plex

text: region

text: of

text: H-II

text: associated

text: with

text: ζ-Ophiuchi,

text: a

text: mas-

text: BWM

text: rates

text: and

text: amplitudes.

text: In

text: Section

text: we

text: summa-

text: sive,

text: runaway

text: O-type

text: star

text: spinning

text: very

text: near

text: breakup

text: rize

text: our

text: key

text: results

text: and

text: oﬀer

text: concluding

text: remarks.

text: We

text: (Gaustad

text: et

text: al.

text: Villamariz

text: Herrero

text: This

text: provide

text: a

text: summary

text: of

text: our

text: important

text: notational

text: elements

text: intervening

text: H-II

text: region

text: could

text: conceivably

text: contribute

text: to

text: in

text: a

text: table

text: of

text: key

text: symbols

text: in

text: an

text: appendix.

text: non-trivial

text: and

text: currently

text: unaccounted-for

text: chromatic

text: ef-

text: PULSAR

text: TIMING

text: DATA

text: SET

text: fects

text: on

text: the

text: timing

text: behavior

text: of

text: In

text: this

text: section,

text: we

text: discuss

text: several

text: aspects

text: of

text: the

text: data

text: SIGNAL

text: MODEL

text: that

text: are

text: relevant

text: to

text: our

text: analysis.

text: For

text: a

text: more

text: thorough

text: For

text: a

text: given

text: pulsar,

text: the

text: timing

text: perturbation

text: from

text: a

text: description

text: of

text: the

text: data,

text: see

text: Demorest

text: and

text: Demor-

text: BWM

text: of

text: amplitude

text: h

text: is

text: well-modeled

text: as

text: B

text: est

text: et

text: al.

text: The

text: ﬁve-year

text: data

text: set

text: consists

text: of

text: pulse

text: times-of-arrival

text: ∆t(t)

text: h

text: B(θ,

text: φ)

text: [(t

text: t

text: )Θ(t

text: t

text: (TOAs)

text: collected

text: at

text: approximately

text: monthly

text: intervals

text: for

text: B

text: each

text: of

text: millisecond

text: pulsars

text: (MSPs).

text: All

text: observations

text: (t

text: t

text: )Θ(t

text: t

text: were

text: done

text: with

text: the

text: Arecibo

text: radio

text: telescope

text: and

text: the

text: Green

text: Bank

text: Telescope

text: (GBT).

text: Of

text: these

text: MSPs,

text: those

text: visi-

text: The

text: function

text: B(θ,

text: φ)

text: cos

text: θ)

text: cos

text: ranges

text: ble

text: to

text: Arecibo

text: were

text: observed

text: with

text: Arecibo;

text: all

text: others

text: between

text: and

text: and

text: is

text: common

text: to

text: all

text: pulsar

text: timing

text: were

text: observed

text: with

text: the

text: GBT.

text: One

text: pulsar,

text: eﬀorts

text: to

text: detect

text: point-like

text: sources

text: of

text: GWs

text: (Estabrook

text: was

text: observed

text: with

text: both

text: telescopes.

text: All

text: observations

text: were

text: Wahlquist

text: Hellings

text: Downs

text: Lee

text: et

text: al.

text: done

text: using

text: one

text: of

text: two

text: identical

text: backend

text: systems:

text: the

text: As-

text: The

text: angle

text: between

text: the

text: direction

text: the

text: burst

text: prop-

text: tronomical

text: Signal

text: Processor

text: (ASP)

text: and

text: the

text: Green

text: Bank

text: agates

text: and

text: the

text: line

text: of

text: sight

text: from

text: Earth

text: to

text: the

text: pulsar

text: is

text: Astronomical

text: Signal

text: Processor

text: (GASP).

text: These

text: backends

text: θ;

text: φ

text: is

text: the

text: angle

text: between

text: the

text: principal

text: polarization

text: vec-

text: performed

text: real-time

text: coherent

text: dedispersion

text: over

text: bands

text: up

text: tor

text: of

text: the

text: wave

text: and

text: the

text: projection

text: of

text: the

text: line

text: of

text: sight

text: to

text: MHz

text: wide

text: and

text: recorded

text: the

text: results

text: averaged

text: over

text: from

text: the

text: Earth

text: to

text: the

text: pulsar

text: onto

text: the

text: plane

text: normal

text: to

text: channels

text: of

text: width

text: MHz

text: each.

text: the

text: wave

text: propagation

text: direction.

text: The

text: BWM

text: encounters

text: For

text: each

text: observing

text: epoch,

text: several

text: TOAs

text: are

text: reported

text: the

text: Earth

text: at

text: a

text: time

text: t

text: and

text: is

text: observed

text: from

text: Earth

text: to

text: from

text: various

text: frequency

text: channels

text: of

text: the

text: MHz

text: band.

text: encounter

text: the

text: pulsar

text: at

text: a

text: time

text: t

text: t

text: cos

text: θ)

text: At

text: Arecibo,

text: observations

text: were

text: typically

text: conducted

text: at

text: where

text: l

text: is

text: the

text: distance

text: from

text: the

text: Earth

text: to

text: the

text: pulsar

text: (van

text: two

text: widely-separated

text: frequencies

text: (usually

text: MHz

text: and

text: Haasteren

text: Levin

text: Cordes

text: Jenet

text: Madison

text: MHz)

text: within

text: one

text: day;

text: at

text: the

text: GBT,

text: observations

text: at

text: et

text: al.

text: The

text: function

text: Θ

text: is

text: the

text: Heaviside

text: step

text: func-

text: two

text: frequencies

text: (usually

text: MHz

text: and

text: MHz)

text: were

text: tion.

text: The

text: amplitude

text: of

text: a

text: BWM

text: coming

text: from

text: a

text: SMBHB

text: conducted

text: within

text: several

text: days

text: of

text: each

text: other.

text: Approx-

text: merger

text: of

text: reduced

text: mass

text: µ

text: M

text: M

text: /(M

text: +M

text: M

text: and

text: imately

text: contemporaneous

text: observations

text: at

text: multiple

text: fre-

text: M

text: are

text: the

text: masses

text: of

text: the

text: black

text: holes

text: in

text: the

text: binary)

text: with

text: quencies

text: allow

text: epoch-to-epoch

text: timing

text: ﬂuctuations

text: caused

text: I

text: a

text: typical

text: inclination

text: angle

text: of

text: at

text: a

text: luminosity

text: by

text: changes

text: in

text: dispersion

text: measure

text: (DM)

text: to

text: be

text: accounted

text: distance

text: D

text: from

text: Earth

text: is

text: (Madison

text: et

text: al.

text: for

text: in

text: the

text: timing

text: model

text: ﬁt

text: (Lam

text: et

text: al.

text: L

text: Three

text: of

text: the

text: pulsars

text: comprising

text: the

text: ﬁve-year

text: NANOGrav

text: data

text: set,

text: and

text: do

text: µ

text: Mpc

text: not

text: have

text: suﬃcient

text: dual-frequency

text: coverage

text: to

text: correct

text: hB

text: D

text: for

text: timing

text: errors

text: from

text: variations

text: in

text: DM

text: over

text: time;

text: we

text: L

text: exclude

text: these

text: pulsars

text: from

text: our

text: analysis

text: for

text: this

text: reason.

text: The

text: distances

text: to

text: the

text: pulsars

text: in

text: the

text: NANOGrav

text: array

text: We

text: also

text: exclude

text: the

text: data

text: set

text: for

text: from

text: our

text: are

text: on

text: the

text: order

text: of

text: kiloparsecs.

text: Our

text: timing

text: baseline

text: is

text: analysis

text: because

text: it

text: is

text: comparatively

text: very

text: short,

text: spanning

text: approximately

text: ﬁve

text: years.

text: Unless

text: θ

text: diﬀers

text: from

text: π

text: by

text: less

text: just

text: two

text: years.

text: Searching

text: for

text: BWMs

text: in

text: such

text: short

text: data

text: than

text: degrees

text: for

text: a

text: particular

text: pulsar,

text: we

text: expect

text: that

text: sets

text: is

text: feasible

text: in

text: principle,

text: but

text: we

text: avoid

text: it

text: for

text: two

text: rea-

text: t

text: and

text: t

text: will

text: not

text: both

text: fall

text: within

text: our

text: observing

text: window.

text: sons.

text: First,

text: the

text: minimum

text: detectable

text: BWM

text: amplitude

text: Since

text: we

text: consider

text: only

text: pulsars

text: in

text: our

text: analysis,

text: less

text: in

text: a

text: particular

text: data

text: set

text: scales

text: approximately

text: as

text: T

text: than

text: of

text: the

text: sky

text: is

text: within

text: degrees

text: of

text: one

text: of

text: our

text: pul-

text: where

text: T

text: is

text: the

text: span

text: of

text: the

text: data

text: set

text: (van

text: Haasteren

text: sars.

text: Assuming

text: BWMs

text: occur

text: isotropically,

text: there

text: is

text: a

text: less

text: Levin

text: so

text: we

text: do

text: not

text: anticipate

text: that

text: this

text: data

text: set

text: than

text: chance

text: of

text: both

text: t

text: and

text: t

text: occurring

text: within

text: our

text: will

text: greatly

text: improve

text: our

text: sensitivity

text: to

text: BWMs.

text: Second,

text: ﬁve-year

text: observing

text: span

text: if

text: a

text: BWM

text: occurs

text: at

text: all.

text: And

text: in

text: such

text: short

text: data

text: sets,

text: many

text: timing

text: model

text: parameters

text: if

text: a

text: BWM

text: occurred

text: with

text: a

text: small

text: enough

text: angular

text: sepa-

text: are

text: highly

text: covariant

text: with

text: each

text: other

text: (e.g.,

text: spin

text: and

text: as-

text: ration

text: from

text: a

text: pulsar

text: in

text: our

text: array

text: that

text: we

text: could

text: see

text: the

text: trometric

text: parameters;

text: Madison

text: et

text: al.

text: and

text: with

text: any

text: timing

text: perturbation

text: both

text: turn

text: on

text: at

text: t

text: and

text: turn

text: oﬀ

text: at

text: t

text: BWM

text: signal

text: present.

text: it

text: would

text: only

text: be

text: observed

text: to

text: turn

text: oﬀ

text: in

text: that

text: one

text: pulsar.

text: Finally

text: we

text: exclude

text: the

text: data

text: for

text: from

text: our

text: So,

text: in

text: each

text: of

text: our

text: pulsar

text: timing

text: data

text: sets,

text: we

text: need

text: only

text: analysis

text: because

text: we

text: believe

text: it

text: contains

text: chromatic

text: tim-

text: to

text: look

text: for

text: evidence

text: of

text: timing

text: perturbations

text: of

text: the

text: form

text: ing

text: biases

text: from

text: unaccounted-for

text: phenomenology

text: in

text: the

text: interstellar

text: medium

text: (ISM;

text: see

text: Figure

text: The

text: DM

text: of

text: is

text: approximately

text: pc

text: cm

text: nearly

text: a

text: factor

text: ∆t(t)

text: h

text: (t

text: t

text: )Θ(t

text: t

text: p

text: B

text: B

text: of

text: two

text: greater

text: than

text: any

text: other

text: pulsar

text: in

text: our

text: sample.

text: With

text: high

text: DM

text: pulsars,

text: chromatic

text: timing

text: errors

text: that

text: deviate

text: from

text: the

text: ν−2

text: scaling

text: expected

text: from

text: cold

text: plasma

text: disper-

text: where

text: hp

text: hBB(θ,

text: φ),

text: what

text: we

text: call

text: the

text: projected

text: burst

text: amplitude,

text: and

text: t

text: is

text: either

text: t

text: or

text: t

text: what

text: we

text: call

text: B

text: ##LTLine##

text: ##LTLine##


text: 3 the burst epoch. Bursts arriving at an individual pulsar from (cid:15) , a random error associated with diﬀrac- DISSν,τ only inﬂuence the timing behavior of that pulsar; we re- tive interstellar scintillation (DISS; Cordes 1990; Cordes fer to these as pulsar-term bursts (see Section 5). Bursts & Shannon 2010). In a detailed study of J1713+0747, arriving at the Earth will simultaneously begin to inﬂu- Shannon & Cordes (2012) found that for single pulses, ≈ ± ence the timing behavior of all pulsars in our sample; we σ 27 1 µs, while timing errors from DISS were only a J refer to these as Earth-term bursts (see Section 6). few nanoseconds. A recent 24-hour study of J1713+0747 4. NOISE MODEL (Dolch et al. 2014) conﬁrmed the jitter measurement of Consider TOAs measured using pulse proﬁles obtained Shannon & Cordes (2012). Errors from DISS can be N by synchronously averaging pulses for each of several substantially more important in the timing of high DM radio-frequency channels. The pulse proﬁles are func- pulsars like B1937+21. We expect DISS to be a sub- tions of pulse phase, ϕ, channel frequency, ν, and ob- dominant component of our timing error budget for the serving epoch, τ. A TOA from a particular frequency MSPs we include in our sample, so we ignore it in our channel and observing epoch can be written as analysis. In summary, the noise covariance matrix for a pulsar in our array, with these anticipated sources of noise, can be approximated a(cid:104)s (cid:105) t = t∞ + t + t + (cid:15) + (cid:15) + (cid:15) , ν,τ ,τ DMν,τ Cν,τ S/Nν,τ Jν,τ DISSν,τ (4) (cid:104) (cid:105) ≈ N N C (cid:48) (cid:48) = (cid:15) (cid:15) (cid:48) (cid:48) δ (cid:48) δ (cid:48)σ2 ( ) + σ2( ) , νν ,ττ ν,τ ν ,τ ττ νν S/N J where t∞ is the TOA at inﬁnite frequency, t is the (7) ,τ DM dispersive delay from propagation through ionized inter- stellar plasma, t is an additional, non-dispersive chro- C matic perturbation, such as intrinsic proﬁle evolution where (cid:15) is the sum of all noise inﬂuencing the TOA ν,τ with frequency, pulse broadening from multipath prop- from observing epoch τ and frequency channel ν. agation, and interstellar refraction (Cordes & Shannon 4.1. Empirical Noise Models 2010). If unaccounted for, t and t can lead to sys- DM C In a previous analysis of the ﬁrst ﬁve years of NANOGrav tematic errors in TOA estimates. With multi-frequency data, Arzoumanian et al. (2014) searched for continuous observations at each observing epoch, eﬀects from tDM GWs using a diﬀer(cid:104)ent noise model: (cid:105) are mitigated in the NANOGrav data set. Potential ef- fects from t are combatted by ﬁtting for constant inter- C Q N J channel oﬀsets unique to each pulsar. Unlike tDM and C (cid:48) (cid:48) =δ (cid:48) δ (cid:48)( 2 + E2σ2 ( )) + 2 + t , the ‘(cid:15)’ terms in Equation 4 are random errors from a νν ,ττ ττ νν S/N C variety of noise sources. C (A, γ). (8) red The term (cid:15) is uncorrelated between frequency S/N ν,τ channels and between observing epochs, i.e., Q The terms and E are commonly referred to as (cid:104) (cid:105) N EQUAD (a source of Gaussian white noise with time (cid:15)S/Nν,τ (cid:15)S/Nν(cid:48),τ(cid:48) = σS2/N( )δνν(cid:48)δττ(cid:48), (5) units added in quadrature to radiometer noise) and EFAC (a dimensionless constant multiplier on the antic- ∝ − ∝ N− ipated amount of radiometer noise), respectively. The where σ (S/N) 1 1/2. The quantity σ J S/N S/N term mimics the correlations induced between TOAs is the TOA uncertainty from radiometer noise assuming from diﬀerent frequency channels of the same observ- that the radiometer noise adds to a ﬁxed pulse shape ing epoch by jitter, but qualitatively diﬀers from jitter under the assumptions of matched ﬁltering against a very in that its value does not change to reﬂect the num- high S/N template pulse proﬁle. N ber of pulses, , averaged together to yield a TOA. The second random contribution to t , (cid:15) , is from phase jitter in single pulses. Jitter is hν,iτghlyJν,cτorrelated Most observations in the ﬁve-year NANOGrav dNata set we considered had similar integration times, so does between frequency channels, but is known to decorre- J not ﬂuctuate substantially between epochs and is thus late over widely-separated frequencies; the probability very much like a jitter-induced correlation. Diﬀerent val- distribution function of the phase of single-pulse cen- Q J ues of , E, and were used for each widely separated troids evolves with frequencies similar to pulse proﬁles observing frequency of each pulsar. (Shannon et al. 2014). Jitter noise can be modeled as The Arzoumanian et al. (2014) noise model also in- cluded red noise with a power-law power spectrum (cid:104) (cid:105) N ≈ N − (cid:15)Jν,τ (cid:15)Jν(cid:48),τ(cid:48) = σJ2( )δττ(cid:48)ρJν,ν(cid:48) σJ2( )δττ(cid:48), (6) Pm(ofd)el=isAn[oft/(t1o ybre t1h)]oγu.ghTthoisf acosmppaortneonftthofe tThOe Anoeisre- ∝ N− ror budget; TOAs measured with extreme accuracy may where σJ 1/2. The(cid:48) quantity ρJν,ν(cid:48) is approxi- still diﬀer from the expectations of a timing model be- mately unity u(cid:46)nless ν and ν are very widely separated. cause of spin noise intrinsic to a pulsar, which is well Usually, σJ σS/N. Jitter noise begins to dominate modeled as a stochastic process with a power-law spec- radiometer noise only when the S/N of single pulses be- trum (Shannon & Cordes 2010). In the Arzoumanian gins to exceed unity. For our data, collected with ASP et al. (2014) analysis, under the assumption that no GW and GASP, jitter noise should be signiﬁcantly subdomi- signal was present in the data set, the noise model pa- Q J nant to radiometer noise for all pulsars. Both radiometer rameters Ξ = [A, γ, E, , ] were determined by ﬁnding and jitter noise are signiﬁcantly larger than contributions the maximum of the likelihood function

label: paragraph

text: the

text: burst

text: epoch.

text: Bursts

text: arriving

text: at

text: an

text: individual

text: pulsar

text: from

text: a

text: random

text: error

text: associated

text: with

text: diﬀrac-

text: DISSν,τ

text: only

text: inﬂuence

text: the

text: timing

text: behavior

text: of

text: that

text: pulsar;

text: we

text: re-

text: tive

text: interstellar

text: scintillation

text: (DISS;

text: Cordes

text: Cordes

text: fer

text: to

text: these

text: as

text: pulsar-term

text: bursts

text: (see

text: Section

text: Bursts

text: Shannon

text: In

text: a

text: detailed

text: study

text: of

text: arriving

text: at

text: the

text: Earth

text: will

text: simultaneously

text: begin

text: to

text: inﬂu-

text: Shannon

text: Cordes

text: found

text: that

text: for

text: single

text: pulses,

text: ence

text: the

text: timing

text: behavior

text: of

text: all

text: pulsars

text: in

text: our

text: sample;

text: we

text: σ

text: µs,

text: while

text: timing

text: errors

text: from

text: DISS

text: were

text: only

text: a

text: J

text: refer

text: to

text: these

text: as

text: Earth-term

text: bursts

text: (see

text: Section

text: few

text: nanoseconds.

text: A

text: recent

text: study

text: of

text: NOISE

text: MODEL

text: (Dolch

text: et

text: al.

text: conﬁrmed

text: the

text: jitter

text: measurement

text: of

text: Consider

text: TOAs

text: measured

text: using

text: pulse

text: proﬁles

text: obtained

text: Shannon

text: Cordes

text: Errors

text: from

text: DISS

text: can

text: be

text: N

text: by

text: synchronously

text: averaging

text: pulses

text: for

text: each

text: of

text: several

text: substantially

text: more

text: important

text: in

text: the

text: timing

text: of

text: high

text: DM

text: radio-frequency

text: channels.

text: The

text: pulse

text: proﬁles

text: are

text: func-

text: pulsars

text: like

text: We

text: expect

text: DISS

text: to

text: be

text: a

text: sub-

text: tions

text: of

text: pulse

text: phase,

text: ϕ,

text: channel

text: frequency,

text: ν,

text: and

text: ob-

text: dominant

text: component

text: of

text: our

text: timing

text: error

text: budget

text: for

text: the

text: serving

text: epoch,

text: τ.

text: A

text: TOA

text: from

text: a

text: particular

text: frequency

text: MSPs

text: we

text: include

text: in

text: our

text: sample,

text: so

text: we

text: ignore

text: it

text: in

text: our

text: channel

text: and

text: observing

text: epoch

text: can

text: be

text: written

text: as

text: analysis.

text: In

text: summary,

text: the

text: noise

text: covariance

text: matrix

text: for

text: a

text: pulsar

text: in

text: our

text: array,

text: with

text: these

text: anticipated

text: sources

text: of

text: noise,

text: can

text: be

text: approximated

text: t

text: t∞

text: t

text: t

text: ν,τ

text: ,τ

text: DMν,τ

text: Cν,τ

text: S/Nν,τ

text: Jν,τ

text: DISSν,τ

text: N

text: N

text: C

text: δ

text: δ

text: νν

text: ,ττ

text: ν,τ

text: ν

text: ,τ

text: ττ

text: νν

text: S/N

text: J

text: where

text: t∞

text: is

text: the

text: TOA

text: at

text: inﬁnite

text: frequency,

text: t

text: is

text: the

text: ,τ

text: DM

text: dispersive

text: delay

text: from

text: propagation

text: through

text: ionized

text: inter-

text: stellar

text: plasma,

text: t

text: is

text: an

text: additional,

text: non-dispersive

text: chro-

text: C

text: matic

text: perturbation,

text: such

text: as

text: intrinsic

text: proﬁle

text: evolution

text: where

text: is

text: the

text: sum

text: of

text: all

text: noise

text: inﬂuencing

text: the

text: TOA

text: ν,τ

text: with

text: frequency,

text: pulse

text: broadening

text: from

text: multipath

text: prop-

text: from

text: observing

text: epoch

text: τ

text: and

text: frequency

text: channel

text: ν.

text: agation,

text: and

text: interstellar

text: refraction

text: (Cordes

text: Shannon

text: Empirical

text: Noise

text: Models

text: If

text: unaccounted

text: for,

text: t

text: and

text: t

text: can

text: lead

text: to

text: sys-

text: DM

text: C

text: In

text: a

text: previous

text: analysis

text: of

text: the

text: ﬁrst

text: ﬁve

text: years

text: of

text: NANOGrav

text: tematic

text: errors

text: in

text: TOA

text: estimates.

text: With

text: multi-frequency

text: data,

text: Arzoumanian

text: et

text: al.

text: searched

text: for

text: continuous

text: observations

text: at

text: each

text: observing

text: epoch,

text: eﬀects

text: from

text: tDM

text: GWs

text: using

text: a

text: noise

text: model:

text: are

text: mitigated

text: in

text: the

text: NANOGrav

text: data

text: set.

text: Potential

text: ef-

text: fects

text: from

text: t

text: are

text: combatted

text: by

text: ﬁtting

text: for

text: constant

text: inter-

text: C

text: Q

text: N

text: J

text: channel

text: oﬀsets

text: unique

text: to

text: each

text: pulsar.

text: Unlike

text: tDM

text: and

text: C

text: =δ

text: δ

text: t

text: the

text: terms

text: in

text: Equation

text: are

text: random

text: errors

text: from

text: a

text: νν

text: ,ττ

text: ττ

text: νν

text: S/N

text: C

text: variety

text: of

text: noise

text: sources.

text: C

text: (A,

text: γ).

text: red

text: The

text: term

text: is

text: uncorrelated

text: between

text: frequency

text: S/N

text: ν,τ

text: channels

text: and

text: between

text: observing

text: epochs,

text: i.e.,

text: Q

text: The

text: terms

text: and

text: E

text: are

text: commonly

text: referred

text: to

text: as

text: N

text: EQUAD

text: (a

text: source

text: of

text: Gaussian

text: white

text: noise

text: with

text: time

text: units

text: added

text: in

text: quadrature

text: to

text: radiometer

text: noise)

text: and

text: EFAC

text: (a

text: dimensionless

text: constant

text: multiplier

text: on

text: the

text: antic-

text: N−

text: ipated

text: amount

text: of

text: radiometer

text: noise),

text: respectively.

text: The

text: where

text: σ

text: (S/N)

text: The

text: quantity

text: σ

text: J

text: S/N

text: S/N

text: term

text: mimics

text: the

text: correlations

text: induced

text: between

text: TOAs

text: is

text: the

text: TOA

text: uncertainty

text: from

text: radiometer

text: noise

text: assuming

text: from

text: diﬀerent

text: frequency

text: channels

text: of

text: the

text: same

text: observ-

text: that

text: the

text: radiometer

text: noise

text: adds

text: to

text: a

text: ﬁxed

text: pulse

text: shape

text: ing

text: epoch

text: by

text: jitter,

text: but

text: qualitatively

text: diﬀers

text: from

text: jitter

text: under

text: the

text: assumptions

text: of

text: matched

text: ﬁltering

text: against

text: a

text: very

text: in

text: that

text: its

text: value

text: does

text: not

text: change

text: to

text: reﬂect

text: the

text: num-

text: high

text: S/N

text: template

text: pulse

text: proﬁle.

text: N

text: ber

text: of

text: pulses,

text: averaged

text: together

text: to

text: yield

text: a

text: TOA.

text: The

text: second

text: random

text: contribution

text: to

text: t

text: is

text: from

text: phase

text: jitter

text: in

text: single

text: pulses.

text: Jitter

text: is

text: hν,iτghlyJν,cτorrelated

text: Most

text: observations

text: in

text: the

text: ﬁve-year

text: NANOGrav

text: dNata

text: set

text: we

text: considered

text: had

text: similar

text: integration

text: times,

text: so

text: does

text: between

text: frequency

text: channels,

text: but

text: is

text: known

text: to

text: decorre-

text: J

text: not

text: ﬂuctuate

text: substantially

text: between

text: epochs

text: and

text: is

text: thus

text: late

text: over

text: widely-separated

text: frequencies;

text: the

text: probability

text: very

text: much

text: like

text: a

text: jitter-induced

text: correlation.

text: Diﬀerent

text: val-

text: distribution

text: function

text: of

text: the

text: phase

text: of

text: single-pulse

text: cen-

text: Q

text: J

text: ues

text: of

text: E,

text: and

text: were

text: used

text: for

text: each

text: widely

text: separated

text: troids

text: evolves

text: with

text: frequencies

text: similar

text: to

text: pulse

text: proﬁles

text: observing

text: frequency

text: of

text: each

text: pulsar.

text: (Shannon

text: et

text: al.

text: Jitter

text: noise

text: can

text: be

text: modeled

text: as

text: The

text: Arzoumanian

text: et

text: al.

text: noise

text: model

text: also

text: in-

text: cluded

text: red

text: noise

text: with

text: a

text: power-law

text: power

text: spectrum

text: N

text: N

text: ybre

text: acosmppaortneonftthofe

text: tThOe

text: Anoeisre-

text: N−

text: ror

text: budget;

text: TOAs

text: measured

text: with

text: extreme

text: accuracy

text: may

text: where

text: σJ

text: quantity

text: is

text: approxi-

text: still

text: diﬀer

text: from

text: the

text: expectations

text: of

text: a

text: timing

text: model

text: be-

text: mately

text: unity

text: ν

text: and

text: ν

text: are

text: very

text: widely

text: separated.

text: cause

text: of

text: spin

text: noise

text: intrinsic

text: to

text: a

text: pulsar,

text: which

text: is

text: well

text: Usually,

text: σJ

text: σS/N.

text: Jitter

text: noise

text: begins

text: to

text: dominate

text: modeled

text: as

text: a

text: stochastic

text: process

text: with

text: a

text: power-law

text: spec-

text: radiometer

text: noise

text: only

text: when

text: the

text: S/N

text: of

text: single

text: pulses

text: be-

text: trum

text: (Shannon

text: Cordes

text: In

text: the

text: Arzoumanian

text: gins

text: to

text: exceed

text: unity.

text: For

text: our

text: data,

text: collected

text: with

text: ASP

text: et

text: al.

text: analysis,

text: under

text: the

text: assumption

text: that

text: no

text: GW

text: and

text: GASP,

text: jitter

text: noise

text: should

text: be

text: signiﬁcantly

text: subdomi-

text: signal

text: was

text: present

text: in

text: the

text: data

text: set,

text: the

text: noise

text: model

text: pa-

text: Q

text: J

text: nant

text: to

text: radiometer

text: noise

text: for

text: all

text: pulsars.

text: Both

text: radiometer

text: rameters

text: Ξ

text: [A,

text: γ,

text: E,

text: were

text: determined

text: by

text: ﬁnding

text: and

text: jitter

text: noise

text: are

text: signiﬁcantly

text: larger

text: than

text: contributions

text: the

text: maximum

text: of

text: the

text: likelihood

text: function


text: 6 epochs to test. With N ﬁxed at ﬁve, inverting Equa- I tion 19 allows us to compute Dˆ for any desired al- thresh lowable false alarm probability; false alarm probabilities of approximately ﬁve and one percent are expected for Dˆ equal to 6.60 and 9.54, respectively. thresh 6. SEARCHES FOR BWMS IN THE EARTH TERM Searches for BWMs in the Earth term have several ad- vantages over searches in pulsar terms. The timing per- turbation from a BWM will turn on simultaneously for all pulsars in the PTA if it occurs in the Earth term, provid- ing a powerful means by which pulsar-speciﬁc phenomena (e.g., glitches) can be ruled out. Variations in the pro- jected BWM amplitude from pulsar to pulsar provide information about the location of the GW source and its polarization. As such, for a trial BWM source location, polarization, and epoch, the residuals of all pulsars in the PTA can be combined into a coherent global ﬁt for the true burst amplitude, h . Such global ﬁts, described in B Madison et al. (2014), have better amplitude sensitivity Fig. 3.— Cumulative distributions of Dˆmax values from 1000 than what is attainable with any one timing data set and simulations of noise-like timing residuals for three of the pulsars in can be carried out with a similar least-squares apparatus our sample. Overlaid are the anticipated cumulative distributions if NI, the eﬀective number of trial burst epochs tested, is one, ﬁve, or as is used in pulsar term searches. twenty. For all pulsars in our sample, the analytically anticipated Global least-squares ﬁtting techniques were recently cumulative distribution best ﬁts the results of our simulations with used by the PPTA to search for BWMs (Wang et al. NI between 4 and 5. 2015). The PPTA data set those authors analyzed con- tained many fewer TOAs than the NANOGrav data set The cumulative distribution for Dˆ associated with we are considering here owing to NANOGrav’s practice (cid:18)(cid:113) max (cid:19) of reporting many TOAs from a single observing epoch f is then simply NI but from diﬀerent frequency channels. With these multi- frequency TOAs, NANOGrav includes as part of its tim- F (Dˆ ) = erfNI Dˆ /2 . (18) ing models numerous chromatic parameters to account NI max max for proﬁle evolution across our wide-bandwidth receivers and time-variable DM, so, we ﬁt for many more timing The anticipated false alarm probability for noise alone model parameters than the PPTA. to exceed a threshold value of Dˆ , Dˆ , is just Suppose N is the number of timing model parame- max thresh ters being ﬁt for each of M pulsars in a PTA (in reality, F − N varies from pulsar to pulsar). Then carrying out a (Dˆ ) = 1 F (Dˆ ). (19) global ﬁt for the amplitude of a BWM requires the in- NI thresh NI thresh × version of an (NM + 1) (NM + 1) matrix, a proce- ∝ For a ﬁxed allowable false alarm probability, Dˆ dure that requires computation time (NM +1)3. This (cid:29) thresh global ﬁt has to be done over N N N trials in the four- grows logarithmically with N if N 1. Ω ψ t I I dimensional phase space of sky position, burst epoch, and To estimate N , for each pulsar, we generated 1000 I burst polarization (N , N , and N are, respectively, the simulated sets of TOAs that matched the real data set in Ω ψ t number of sky positions, polarization angles, and burst number of TOAs, observing schedule, and timing model, epochs tested). Methods requiring these global ﬁts are but yielded timing residuals consistent with our noise thus computationally onerous for NANOGrav because of models. We then ﬁt for hˆ (t ) along an equispaced grid p B the large number of timing model parameters for which of twenty trial burst epochs between MJD 53500 and we ﬁt. More importantly, the total number of suitably 54900 and computed Dˆ(tB) to get Dˆmax. stable MSPs being timed by NANOGrav is growing with In Figure 3, we show the cumulative distribution of time as projects like the Arecibo PALFA survey (e.g., the 1000 Dˆ values from our simulations for three pul- Cordes et al. 2006; Swiggum et al. 2014), the Arecibo max − sars: J0030+0451 and J2145 0750, observed by Arecibo All-sky 327-MHz Drift Pulsar Survey (e.g., Deneva et al. and the GBT, respectively, with rms timing residuals of 2013), and the GBT Northern Celestial Cap Pulsar Sur- ∼ 100 ns, and J1713+0747, observed by both Arecibo vey (e.g., Stovall et al. 2014) continue to ﬁnd more pul- ∼ and the GBT with an rms timing residual of 30 ns. sars. A search procedure based on global ﬁts will not We also plot the theoretically anticipated curves for N scale well to future PTA endeavors. We use a diﬀerent I equal to one, ﬁve, and twenty. In ﬁtting the anticipated technique that avoids having to carry out any global ﬁts cumulative distributions of Dˆ with N as a free pa- and is computationally much more eﬃcient. max I rameter to the results of our simulations, we ﬁnd that for 6.1. Accelerated Earth-term Searches all pulsars, NI is between four and ﬁve. As NI increases, Over a ﬁve-dimensional grid of trial burst sky posi- large values of Dˆ occur more frequently in the pres- tions, Ω , polarizations, ψ , epochs, t , and ampli- max ij k B,l ence of pure noise. We take the conservative approach tudes, h , we search for the maximum in the global B,m and assume that there are ﬁve independent trial burst ##LTFigure## ##LTLine##

text: epochs

text: to

text: test.

text: With

text: N

text: ﬁxed

text: at

text: ﬁve,

text: inverting

text: Equa-

text: I

text: tion

text: allows

text: us

text: to

text: compute

text: Dˆ

text: for

text: any

text: desired

text: al-

text: thresh

text: lowable

text: false

text: alarm

text: probability;

text: false

text: alarm

text: probabilities

text: of

text: approximately

text: ﬁve

text: and

text: one

text: percent

text: are

text: expected

text: for

text: Dˆ

text: equal

text: to

text: and

text: respectively.

text: thresh

label: section

text: SEARCHES

text: FOR

text: BWMS

text: IN

text: THE

text: EARTH

text: TERM

text: Searches

text: for

text: BWMs

text: in

text: the

text: Earth

text: term

text: have

text: several

text: ad-

text: vantages

text: over

text: searches

text: in

text: pulsar

text: terms.

text: The

text: timing

text: per-

text: turbation

text: from

text: a

text: BWM

text: will

text: turn

text: on

text: simultaneously

text: for

text: all

text: pulsars

text: in

text: the

text: PTA

text: if

text: it

text: occurs

text: in

text: the

text: Earth

text: term,

text: provid-

text: ing

text: a

text: powerful

text: means

text: by

text: which

text: pulsar-speciﬁc

text: phenomena

text: (e.g.,

text: glitches)

text: can

text: be

text: ruled

text: out.

text: Variations

text: in

text: the

text: pro-

text: jected

text: BWM

text: amplitude

text: from

text: pulsar

text: to

text: pulsar

text: provide

text: information

text: about

text: the

text: location

text: of

text: the

text: GW

text: source

text: and

text: its

text: polarization.

text: As

text: such,

text: for

text: a

text: trial

text: BWM

text: source

text: location,

text: polarization,

text: and

text: epoch,

text: the

text: residuals

text: of

text: all

text: pulsars

text: in

text: the

text: PTA

text: can

text: be

text: combined

text: into

text: a

text: coherent

text: global

text: ﬁt

text: for

text: the

text: true

text: burst

text: amplitude,

text: h

text: Such

text: global

text: ﬁts,

text: described

text: in

text: B

text: Madison

text: et

text: al.

text: have

text: better

text: amplitude

text: sensitivity

text: Fig.

text: Cumulative

text: distributions

text: of

text: Dˆmax

text: values

text: from

text: than

text: what

text: is

text: attainable

text: with

text: any

text: one

text: timing

text: data

text: set

text: and

text: simulations

text: of

text: noise-like

text: timing

text: residuals

text: for

text: three

text: of

text: the

text: pulsars

text: in

text: can

text: be

text: carried

text: out

text: with

text: a

text: similar

text: least-squares

text: apparatus

text: our

text: sample.

text: Overlaid

text: are

text: the

text: anticipated

text: cumulative

text: distributions

text: if

text: NI,

text: the

text: eﬀective

text: number

text: of

text: trial

text: burst

text: epochs

text: tested,

text: is

text: one,

text: ﬁve,

text: or

text: as

text: is

text: used

text: in

text: pulsar

text: term

text: searches.

text: twenty.

text: For

text: all

text: pulsars

text: in

text: our

text: sample,

text: the

text: analytically

text: anticipated

text: Global

text: least-squares

text: ﬁtting

text: techniques

text: were

text: recently

text: cumulative

text: distribution

text: best

text: ﬁts

text: the

text: results

text: of

text: our

text: simulations

text: with

text: used

text: by

text: the

text: PPTA

text: to

text: search

text: for

text: BWMs

text: (Wang

text: et

text: al.

text: NI

text: between

text: and

text: The

text: PPTA

text: data

text: set

text: those

text: authors

text: analyzed

text: con-

text: tained

text: many

text: fewer

text: TOAs

text: than

text: the

text: NANOGrav

text: data

text: set

text: The

text: cumulative

text: distribution

text: for

text: Dˆ

text: associated

text: with

text: we

text: are

text: considering

text: here

text: owing

text: to

text: NANOGrav’s

text: practice

text: max

text: of

text: reporting

text: many

text: TOAs

text: from

text: a

text: single

text: observing

text: epoch

text: f

text: is

text: then

text: simply

text: NI

text: but

text: from

text: diﬀerent

text: frequency

text: channels.

text: With

text: these

text: multi-

text: frequency

text: TOAs,

text: NANOGrav

text: includes

text: as

text: part

text: of

text: its

text: tim-

text: F

text: (Dˆ

text: erfNI

text: Dˆ

text: ing

text: models

text: numerous

text: chromatic

text: parameters

text: to

text: account

text: NI

text: max

text: max

text: for

text: proﬁle

text: evolution

text: across

text: our

text: wide-bandwidth

text: receivers

text: and

text: time-variable

text: DM,

text: so,

text: we

text: ﬁt

text: for

text: many

text: more

text: timing

text: The

text: anticipated

text: false

text: alarm

text: probability

text: for

text: noise

text: alone

text: model

text: parameters

text: than

text: the

text: PPTA.

text: to

text: exceed

text: a

text: threshold

text: value

text: of

text: Dˆ

text: Dˆ

text: is

text: just

text: Suppose

text: N

text: is

text: the

text: number

text: of

text: timing

text: model

text: parame-

text: max

text: thresh

text: ters

text: being

text: ﬁt

text: for

text: each

text: of

text: M

text: pulsars

text: in

text: a

text: PTA

text: (in

text: reality,

text: F

text: N

text: varies

text: from

text: pulsar

text: to

text: pulsar).

text: Then

text: carrying

text: out

text: a

text: (Dˆ

text: F

text: (Dˆ

text: global

text: ﬁt

text: for

text: the

text: amplitude

text: of

text: a

text: BWM

text: requires

text: the

text: in-

text: NI

text: thresh

text: NI

text: thresh

text: version

text: of

text: an

text: (NM

text: (NM

text: matrix,

text: a

text: proce-

text: For

text: a

text: ﬁxed

text: allowable

text: false

text: alarm

text: probability,

text: Dˆ

text: dure

text: that

text: requires

text: computation

text: time

text: (NM

text: This

text: thresh

text: global

text: ﬁt

text: has

text: to

text: be

text: done

text: over

text: N

text: N

text: N

text: trials

text: in

text: the

text: four-

text: grows

text: logarithmically

text: with

text: N

text: if

text: N

text: Ω

text: ψ

text: t

text: I

text: I

text: dimensional

text: phase

text: space

text: of

text: sky

text: position,

text: burst

text: epoch,

text: and

text: To

text: estimate

text: N

text: for

text: each

text: pulsar,

text: we

text: generated

text: I

text: burst

text: polarization

text: (N

text: N

text: and

text: N

text: are,

text: respectively,

text: the

text: simulated

text: sets

text: of

text: TOAs

text: that

text: matched

text: the

text: real

text: data

text: set

text: in

text: Ω

text: ψ

text: t

text: number

text: of

text: sky

text: positions,

text: polarization

text: angles,

text: and

text: burst

text: number

text: of

text: TOAs,

text: observing

text: schedule,

text: and

text: timing

text: model,

text: epochs

text: tested).

text: Methods

text: requiring

text: these

text: global

text: ﬁts

text: are

text: but

text: yielded

text: timing

text: residuals

text: consistent

text: with

text: our

text: noise

text: thus

text: computationally

text: onerous

text: for

text: NANOGrav

text: because

text: of

text: models.

text: We

text: then

text: ﬁt

text: for

text: hˆ

text: (t

text: along

text: an

text: equispaced

text: grid

text: p

text: B

text: the

text: large

text: number

text: of

text: timing

text: model

text: parameters

text: for

text: which

text: of

text: twenty

text: trial

text: burst

text: epochs

text: between

text: MJD

text: and

text: we

text: ﬁt.

text: More

text: importantly,

text: the

text: total

text: number

text: of

text: suitably

text: and

text: computed

text: Dˆ(tB)

text: to

text: get

text: Dˆmax.

text: stable

text: MSPs

text: being

text: timed

text: by

text: NANOGrav

text: is

text: growing

text: with

text: In

text: Figure

text: we

text: show

text: the

text: cumulative

text: distribution

text: of

text: time

text: as

text: projects

text: like

text: the

text: Arecibo

text: PALFA

text: survey

text: (e.g.,

text: the

text: Dˆ

text: values

text: from

text: our

text: simulations

text: for

text: three

text: pul-

text: Cordes

text: et

text: al.

text: Swiggum

text: et

text: al.

text: the

text: Arecibo

text: max

text: sars:

text: and

text: observed

text: by

text: Arecibo

text: All-sky

text: Drift

text: Pulsar

text: Survey

text: (e.g.,

text: Deneva

text: et

text: al.

text: and

text: the

text: GBT,

text: respectively,

text: with

text: rms

text: timing

text: residuals

text: of

text: and

text: the

text: GBT

text: Northern

text: Celestial

text: Cap

text: Pulsar

text: Sur-

text: ns,

text: and

text: observed

text: by

text: both

text: Arecibo

text: vey

text: (e.g.,

text: Stovall

text: et

text: al.

text: continue

text: to

text: ﬁnd

text: more

text: pul-

text: and

text: the

text: GBT

text: with

text: an

text: rms

text: timing

text: residual

text: of

text: ns.

text: sars.

text: A

text: search

text: procedure

text: based

text: on

text: global

text: ﬁts

text: will

text: not

text: We

text: also

text: plot

text: the

text: theoretically

text: anticipated

text: curves

text: for

text: N

text: scale

text: well

text: to

text: future

text: PTA

text: endeavors.

text: We

text: use

text: a

text: diﬀerent

text: I

text: equal

text: to

text: one,

text: ﬁve,

text: and

text: twenty.

text: In

text: ﬁtting

text: the

text: anticipated

text: technique

text: that

text: avoids

text: having

text: to

text: carry

text: out

text: any

text: global

text: ﬁts

text: cumulative

text: distributions

text: of

text: Dˆ

text: with

text: N

text: as

text: a

text: free

text: pa-

text: and

text: is

text: computationally

text: much

text: more

text: eﬃcient.

text: max

text: I

text: rameter

text: to

text: the

text: results

text: of

text: our

text: simulations,

text: we

text: ﬁnd

text: that

text: for

text: Accelerated

text: Earth-term

text: Searches

text: all

text: pulsars,

text: NI

text: is

text: between

text: four

text: and

text: ﬁve.

text: As

text: NI

text: increases,

text: Over

text: a

text: ﬁve-dimensional

text: grid

text: of

text: trial

text: burst

text: sky

text: posi-

text: large

text: values

text: of

text: Dˆ

text: occur

text: more

text: frequently

text: in

text: the

text: pres-

text: tions,

text: Ω

text: polarizations,

text: ψ

text: epochs,

text: t

text: and

text: ampli-

text: max

text: ij

text: k

text: B,l

text: ence

text: of

text: pure

text: noise.

text: We

text: take

text: the

text: conservative

text: approach

text: tudes,

text: h

text: we

text: search

text: for

text: the

text: maximum

text: in

text: the

text: global

text: B,m

text: and

text: assume

text: that

text: there

text: are

text: ﬁve

text: independent

text: trial

text: burst

text: ##LTFigure##

text: ##LTLine##


text: 8 ing residuals outside of the window of tested trial burst TABLE 1 epochs, when testing the ﬁrst or last trial epoch, the Summary of Results from Pulsar Term BWM Search pre- or post-burst timing model is constrained by a − F small number of data points. If the residuals at the edge PSR hˆmp ax/10 13 tmB,aix Dˆmax log10 ( 5) of the data set are slight outliers, allowing for an instan- (MJD) taneous change in the spin period of the pulsar near the − ± − J0030−+0451 0.87±0.77 54100 1.2 −0.11 beginning or end of the data set can bring them more in J0613 0200 0.68±0.50 54584 1.8 −0.20 line with zero and lead to a modest reduction of the χ2 JJ11041525−+35333007 −−21..5477±±30..3759 5544206112 30..54 −−00..5042 value of the residuals. J1640+2224 2.65±1.45 53801 3.3 −0.52 The individual data set in our sample that most tightly J1713−+0747 −0.08±0.07 54808 1.4 −0.13 constrains BWMs is for J1713+0747. This is in line with J1744 1134 2.06±1.19 53541 2.9 −0.45 the expectations of Madison et al. (2014). In Figure 4, we B1855−+09 −1.00±0.73 53578 1.8 −0.21 J1909−3744 −1.29±0.74 54994 3.0 −0.45 show the detailed BWM search results for J1713+0747. J1918−0642 8.29±3.39 54994 5.9 −1.15 The bottom panel is simply the epoch-averaged residuals J2145 0750 −26.60±11.51 54994 5.3 −0.99 as in Figure 2; we plot them again here to explicitly show J2317+1439 2.40 1.20 53541 3.9 0.67 how the interval of trial burst epochs tested overlaps with Note. — Description of most signiﬁcant BWM-like signal de- the TOA coverage. The second panel from the bottom tected in the pulsar term of each of 12 NANOGrav data sets. From shows hˆ ± σ at the 40 trial burst epochs we tested. left to right, the columns are the name of the pulsar, the most sig- p h,p niﬁcant projected BWM amplitude calculated from least-squares The best-ﬁt projected amplitude is completely consistent ﬁtting, the trial burst epoch at which the most signiﬁcant ampli- with zero at each trial burst epoch considered. Further tude was found, the corresponding Dˆmax value, and the logarithm echoing this, in the third panel from the bottom, we show of the anticipated false alarm probability for that value of Dˆmax the value of Dˆ as it varies with trial burst epoch. Never aepssoucmhsintgestNeId,, itshﬁevne.umber of eﬀectively independent trial burst does Dˆ approach the values necessary to be inconsistent with noise at the ﬁve- or one-percent level. In the re- As an independent check, we also carry out the Bayesian maining panels of Figure 4, we show a “heat map” of method of van Haasteren & Levin (2010), implemented the two-dimensional likelihood ratio surface, as in Equa- in the software package Piccard21, developed indepen- tion 12, that we use in our Earth-term search. dently of the Madison et al. (2014) method. In brief, 8. EARTH TERM RESULTS this Bayesian method takes the likelihood of Equation In Figure 5, we show a two-dimensional slice of our 9 as a function of the noise parameters Ξ, the tim- Earth-term search results. For ﬁxed trial burst polar- ing model parameters δp, and the BWM parameters (Ωij, ψk, tB,l, hB,m). Similar to what we do for our fre- ization angle and epoch, we depict the quantity D˜G1/2 as quentist search, we keep the noise parameters ﬁxed to it varies with trial burst source location. The quantity the values obtained in Arzoumanian et al. (2014). Our D˜1/2 is the best-ﬁt BWM amplitude, h˜ , in units of the G B prior distributions are ﬂat in all stated BWM parame- uncertainty on the amplitude, σ (see Equation 22). The h ters, where we note that the prior in Ω is taken ﬂat slice we show contains the smallest value of σ found in ij h over the sphere. We have been able to conﬁrm that the our entire search, i.e., the point in our parameter space results we describe in this paper are identical for the where we are most sensitive to a BWM; its location on frequentist and Bayesian methods. the sky is indicated by the black circle very near the lo- 7. PULSAR TERM RESULTS cation of J1713+0747. For all twelve of the pulsar data sets we analyze, our The single largest value of D˜G we ﬁnd is D˜G,max = search for pulsar-term BWMs occurring between MJDs 6.03, which corresponds to h˜ = 2.46σ . We mentioned G h 53541 and 54994 yields results that are entirely consis- at the end of Section 6.1 that the eﬀective number of tent with our noise models. In Table 1, we summarize trials tested in an Earth-term BWM search, N , will G the key results in our search for pulsar-term BWMs. For be larger than N = 5 because of the search over not I each pulsar in our sample, we list the pulsar name, the just many trial burst epochs, but also over many trial most signiﬁcant value of hˆ (according to the Dˆ num- burst source locations and polarizations, implying that p ber) along with its 1-σ uncertainty, the trial burst epoch for a ﬁxed allowable false-alarm probability we have to at which this most signiﬁcant amplitude was found, the raise the threshold we impose on D˜ above the threshold G Dˆ value for that pulsar, and the logarithm of the used on the detection statistic in a pulsar-term search. max false alarm probability anticipated from noise alone for However, D˜ is small enough that even if N were G,max G that value of Dˆ assuming N = 5. The most sig- only ﬁve, this result would be entirely consistent with max I − niﬁcant event we ﬁnd is in the data for J1918 0642; it more than 95% of realizations of our noise. is consistent with approximately seven percent of noise In the BWM analysis carried out by Wang et al. (2015), realizations. when faced with marginally high values of their test The epochs at which the most signiﬁcant projected statistic, they conducted an extensive suite of simula- BWM amplitudes occur are distributed nearly uniformly tions to assess NG, comparable to what we have done throughout the range of trial epochs we tested with to assess NI (and depicted in Figure 3), and were able repeat values occurring only at the very ﬁrst and last to justiﬁably raise the detection threshold on their test epochs tested. Clustering at the edges of the window statistic and rule out a detection. We ﬁnd no comparably of tested epochs is not surprising. If there are few tim- large values of D˜ that exceed our detection threshold G even if we underestimate N as ﬁve. G 21 https://github.com/vhaasteren/piccard We do still want an estimate of N as it will allow us to G ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ing

text: residuals

text: outside

text: of

text: the

text: window

text: of

text: tested

text: trial

text: burst

text: TABLE

text: epochs,

text: when

text: testing

text: the

text: ﬁrst

text: or

text: last

text: trial

text: epoch,

text: the

text: Summary

text: of

text: Results

text: from

text: Pulsar

text: Term

text: BWM

text: Search

text: pre-

text: or

text: post-burst

text: timing

text: model

text: is

text: constrained

text: by

text: a

text: F

text: small

text: number

text: of

text: data

text: points.

text: If

text: the

text: residuals

text: at

text: the

text: edge

text: PSR

text: hˆmp

text: tmB,aix

text: Dˆmax

text: of

text: the

text: data

text: set

text: are

text: slight

text: outliers,

text: allowing

text: for

text: an

text: instan-

text: (MJD)

text: taneous

text: change

text: in

text: the

text: spin

text: period

text: of

text: the

text: pulsar

text: near

text: the

text: beginning

text: or

text: end

text: of

text: the

text: data

text: set

text: can

text: bring

text: them

text: more

text: in

text: line

text: with

text: zero

text: and

text: lead

text: to

text: a

text: modest

text: reduction

text: of

text: the

text: χ2

text: value

text: of

text: the

text: residuals.

text: The

text: individual

text: data

text: set

text: in

text: our

text: sample

text: that

text: most

text: tightly

text: constrains

text: BWMs

text: is

text: for

text: This

text: is

text: in

text: line

text: with

text: the

text: expectations

text: of

text: Madison

text: et

text: al.

text: In

text: Figure

text: we

text: show

text: the

text: detailed

text: BWM

text: search

text: results

text: for

text: The

text: bottom

text: panel

text: is

text: simply

text: the

text: epoch-averaged

text: residuals

text: as

text: in

text: Figure

text: we

text: plot

text: them

text: again

text: here

text: to

text: explicitly

text: show

text: how

text: the

text: interval

text: of

text: trial

text: burst

text: epochs

text: tested

text: overlaps

text: with

text: Note.

text: Description

text: of

text: most

text: signiﬁcant

text: BWM-like

text: signal

text: de-

text: the

text: TOA

text: coverage.

text: The

text: second

text: panel

text: from

text: the

text: bottom

text: tected

text: in

text: the

text: pulsar

text: term

text: of

text: each

text: of

text: NANOGrav

text: data

text: sets.

text: From

text: shows

text: hˆ

text: σ

text: at

text: the

text: trial

text: burst

text: epochs

text: we

text: tested.

text: left

text: to

text: right,

text: the

text: columns

text: are

text: the

text: name

text: of

text: the

text: pulsar,

text: the

text: most

text: sig-

text: p

text: h,p

text: niﬁcant

text: projected

text: BWM

text: amplitude

text: calculated

text: from

text: least-squares

text: The

text: best-ﬁt

text: projected

text: amplitude

text: is

text: completely

text: consistent

text: ﬁtting,

text: the

text: trial

text: burst

text: epoch

text: at

text: which

text: the

text: most

text: signiﬁcant

text: ampli-

text: with

text: zero

text: at

text: each

text: trial

text: burst

text: epoch

text: considered.

text: Further

text: tude

text: was

text: found,

text: the

text: corresponding

text: Dˆmax

text: value,

text: and

text: the

text: logarithm

text: echoing

text: this,

text: in

text: the

text: third

text: panel

text: from

text: the

text: bottom,

text: we

text: show

text: of

text: the

text: anticipated

text: false

text: alarm

text: probability

text: for

text: that

text: value

text: of

text: Dˆmax

text: the

text: value

text: of

text: Dˆ

text: as

text: it

text: varies

text: with

text: trial

text: burst

text: epoch.

text: Never

text: aepssoucmhsintgestNeId,,

text: itshﬁevne.umber

text: of

text: eﬀectively

text: independent

text: trial

text: burst

text: does

text: Dˆ

text: approach

text: the

text: values

text: necessary

text: to

text: be

text: inconsistent

text: with

text: noise

text: at

text: the

text: ﬁve-

text: or

text: one-percent

text: level.

text: In

text: the

text: re-

text: As

text: an

text: independent

text: check,

text: we

text: also

text: carry

text: out

text: the

text: Bayesian

text: maining

text: panels

text: of

text: Figure

text: we

text: show

text: a

text: “heat

text: map”

text: of

text: method

text: of

text: van

text: Haasteren

text: Levin

text: implemented

text: the

text: two-dimensional

text: likelihood

text: ratio

text: surface,

text: as

text: in

text: Equa-

text: in

text: the

text: software

text: package

text: developed

text: indepen-

text: tion

text: that

text: we

text: use

text: in

text: our

text: Earth-term

text: search.

text: dently

text: of

text: the

text: Madison

text: et

text: al.

text: method.

text: In

text: brief,

text: EARTH

text: TERM

text: RESULTS

text: this

text: Bayesian

text: method

text: takes

text: the

text: likelihood

text: of

text: Equation

text: In

text: Figure

text: we

text: show

text: a

text: two-dimensional

text: slice

text: of

text: our

text: as

text: a

text: function

text: of

text: the

text: noise

text: parameters

text: Ξ,

text: the

text: tim-

text: Earth-term

text: search

text: results.

text: For

text: ﬁxed

text: trial

text: burst

text: polar-

text: ing

text: model

text: parameters

text: δp,

text: and

text: the

text: BWM

text: parameters

text: (Ωij,

text: ψk,

text: tB,l,

text: hB,m).

text: Similar

text: to

text: what

text: we

text: do

text: for

text: our

text: fre-

text: ization

text: angle

text: and

text: epoch,

text: we

text: depict

text: the

text: quantity

text: as

text: quentist

text: search,

text: we

text: keep

text: the

text: noise

text: parameters

text: ﬁxed

text: to

text: it

text: varies

text: with

text: trial

text: burst

text: source

text: location.

text: The

text: quantity

text: the

text: values

text: obtained

text: in

text: Arzoumanian

text: et

text: al.

text: Our

text: is

text: the

text: best-ﬁt

text: BWM

text: amplitude,

text: h˜

text: in

text: units

text: of

text: the

text: G

text: B

text: prior

text: distributions

text: are

text: ﬂat

text: in

text: all

text: stated

text: BWM

text: parame-

text: uncertainty

text: on

text: the

text: amplitude,

text: σ

text: (see

text: Equation

text: The

text: h

text: ters,

text: where

text: we

text: note

text: that

text: the

text: prior

text: in

text: Ω

text: is

text: taken

text: ﬂat

text: slice

text: we

text: show

text: contains

text: the

text: smallest

text: value

text: of

text: σ

text: found

text: in

text: ij

text: h

text: over

text: the

text: sphere.

text: We

text: have

text: been

text: able

text: to

text: conﬁrm

text: that

text: the

text: our

text: entire

text: search,

text: i.e.,

text: the

text: point

text: in

text: our

text: parameter

text: space

text: results

text: we

text: describe

text: in

text: this

text: paper

text: are

text: identical

text: for

text: the

text: where

text: we

text: are

text: most

text: sensitive

text: to

text: a

text: BWM;

text: its

text: location

text: on

text: frequentist

text: and

text: Bayesian

text: methods.

text: the

text: sky

text: is

text: indicated

text: by

text: the

text: black

text: circle

text: very

text: near

text: the

text: lo-

text: PULSAR

text: TERM

text: RESULTS

text: cation

text: of

text: For

text: all

text: twelve

text: of

text: the

text: pulsar

text: data

text: sets

text: we

text: analyze,

text: our

text: The

text: single

text: largest

text: value

text: of

text: D˜G

text: we

text: ﬁnd

text: is

text: D˜G,max

text: search

text: for

text: pulsar-term

text: BWMs

text: occurring

text: between

text: MJDs

text: which

text: corresponds

text: to

text: h˜

text: We

text: mentioned

text: G

text: h

text: and

text: yields

text: results

text: that

text: are

text: entirely

text: consis-

text: at

text: the

text: end

text: of

text: Section

text: that

text: the

text: eﬀective

text: number

text: of

text: tent

text: with

text: our

text: noise

text: models.

text: In

text: Table

text: we

text: summarize

text: trials

text: tested

text: in

text: an

text: Earth-term

text: BWM

text: search,

text: N

text: will

text: G

text: the

text: key

text: results

text: in

text: our

text: search

text: for

text: pulsar-term

text: BWMs.

text: For

text: be

text: larger

text: than

text: N

text: because

text: of

text: the

text: search

text: over

text: not

text: I

text: each

text: pulsar

text: in

text: our

text: sample,

text: we

text: list

text: the

text: pulsar

text: name,

text: the

text: just

text: many

text: trial

text: burst

text: epochs,

text: but

text: also

text: over

text: many

text: trial

text: most

text: signiﬁcant

text: value

text: of

text: hˆ

text: (according

text: to

text: the

text: Dˆ

text: num-

text: burst

text: source

text: locations

text: and

text: polarizations,

text: implying

text: that

text: p

text: ber)

text: along

text: with

text: its

text: uncertainty,

text: the

text: trial

text: burst

text: epoch

text: for

text: a

text: ﬁxed

text: allowable

text: false-alarm

text: probability

text: we

text: have

text: to

text: at

text: which

text: this

text: most

text: signiﬁcant

text: amplitude

text: was

text: found,

text: the

text: raise

text: the

text: threshold

text: we

text: impose

text: on

text: D˜

text: above

text: the

text: threshold

text: G

text: Dˆ

text: value

text: for

text: that

text: pulsar,

text: and

text: the

text: logarithm

text: of

text: the

text: used

text: on

text: the

text: detection

text: statistic

text: in

text: a

text: pulsar-term

text: search.

text: max

text: false

text: alarm

text: probability

text: anticipated

text: from

text: noise

text: alone

text: for

text: However,

text: D˜

text: is

text: small

text: enough

text: that

text: even

text: if

text: N

text: were

text: G,max

text: G

text: that

text: value

text: of

text: Dˆ

text: assuming

text: N

text: The

text: most

text: sig-

text: only

text: ﬁve,

text: this

text: result

text: would

text: be

text: entirely

text: consistent

text: with

text: max

text: I

text: niﬁcant

text: event

text: we

text: ﬁnd

text: is

text: in

text: the

text: data

text: for

text: it

text: more

text: than

text: of

text: realizations

text: of

text: our

text: noise.

text: is

text: consistent

text: with

text: approximately

text: seven

text: percent

text: of

text: noise

text: In

text: the

text: BWM

text: analysis

text: carried

text: out

text: by

text: Wang

text: et

text: al.

text: realizations.

text: when

text: faced

text: with

text: marginally

text: high

text: values

text: of

text: their

text: test

text: The

text: epochs

text: at

text: which

text: the

text: most

text: signiﬁcant

text: projected

text: statistic,

text: they

text: conducted

text: an

text: extensive

text: suite

text: of

text: simula-

text: BWM

text: amplitudes

text: occur

text: are

text: distributed

text: nearly

text: uniformly

text: tions

text: to

text: assess

text: NG,

text: comparable

text: to

text: what

text: we

text: have

text: done

text: throughout

text: the

text: range

text: of

text: trial

text: epochs

text: we

text: tested

text: with

text: to

text: assess

text: NI

text: (and

text: depicted

text: in

text: Figure

text: and

text: were

text: able

text: repeat

text: values

text: occurring

text: only

text: at

text: the

text: very

text: ﬁrst

text: and

text: last

text: to

text: justiﬁably

text: raise

text: the

text: detection

text: threshold

text: on

text: their

text: test

text: epochs

text: tested.

text: Clustering

text: at

text: the

text: edges

text: of

text: the

text: window

text: statistic

text: and

text: rule

text: out

text: a

text: detection.

text: We

text: ﬁnd

text: no

text: comparably

text: of

text: tested

text: epochs

text: is

text: not

text: surprising.

text: If

text: there

text: are

text: few

text: tim-

text: large

text: values

text: of

text: D˜

text: that

text: exceed

text: our

text: detection

text: threshold

text: G

text: even

text: if

text: we

text: underestimate

text: N

text: as

text: ﬁve.

text: G

text: We

text: do

text: still

text: want

text: an

text: estimate

text: of

text: N

text: as

text: it

text: will

text: allow

text: us

text: to

text: G

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 − This has little to do with J0613 0200, but is instead because these sky positions are antipodal to our great- est concentration of pulsars, especially our most sensitive pulsar, J1713+0747. The green triangle in Figure 6 in- dicates the position of the Virgo Cluster. Just 16.5 Mpc from Earth, the Virgo Cluster falls very near the edge of the volume over which we are sensitive to BWMs from I µ = 109 M(cid:12), = π/3 binary mergers. 9. BWM RATE-AMPLITUDE CONSTRAINTS To synthesize the results from both our Earth- and pulsar-term analyses, we derive constraints on Λ(> h ), B the annual rate of BWMs from any part of the sky with any polarization having amplitudes greater than h , as- B suming they occur as a Poisson process. This quantity is readily relatable to astrophysical models of processes producing BWMs, i.e., Equation 15 of Cordes & Jenet (2012), Equation 21 of Madison et al. (2014), or Equation 17 of Wang et al. (2015). Toward this end, deﬁne the quantities, (cid:88) (cid:88) (cid:88) Fig. 7.— 95% conﬁdence Earth-term upper bound on the rate, Λ(> hB), of BWMs occurring with amplitudes at or above ampli- ∆t∆Ω∆ψ Nt NΩ Nψ − tudes hB. The two curves come from our Earth-term and pulsar- τ (h , n)= Θ(h nσ ), term analyses. The pulsar-term probes lower-rate events at high E B 4π2 B h,ijk amplitudes because individual pulsar terms contain causally inde- i j k pendent information. The near total convergence of the Earth- and (24) pulsar-term curves at low BWM amplitudes demonstrates that a single pulsar, J1713+0747, is dominating our sensitivity. and (cid:88) (cid:88) (cid:90) (cid:18) (cid:19) in an Earth-term search with NG = 120. Our estimate of N is likely signiﬁcantly larger than it needs to be. A G detailed suite of simulations could give us a more realistic ∆t M Nt − nσ assessment of N which would lead to more constrain- τ (h , n)= Θ h h,p,K dψdΩ, G P B 4π2 B B(θ, φ) ing upper limits, but for a ﬁxed allowable false alarm K i probability, the amplitude a BWM must exceed to be (25) inconsistent with the noise scales as the square root of the natural logarithm of N , so the improvement to the G upper limit from this analysis would be very slight. where ∆t, ∆Ω, and ∆ψ describe the grid spacing in Since there are fewer eﬀectively independent trials in a the Earth-term search. The quantities τ and τ are the E P pulsar-term search, we should use a lower value of n for a total time that the PTA had n-σ sensitivity to a BWM of 95% conﬁdence upper limit. But, by treating the Earth- amplitude at least h in the Earth-term and pulsar-term, B and pulsar-terms comparably, we are able to highlight respectively, weighted by the fraction of the total source- an important fact about our data set. The near-total location and polarization angle space over which that convergence of our Earth- and pulsar-term constraints sensitivity was achieved. Our deﬁnitions for τ and τ E P for low values of h indicates that our upper limits are diﬀer from nearly identical deﬁnitions in Madison et al. B almost entirely dominated by a single pulsar data set: (2014) by an overall factor of two. In that work, it was J1713+0747. This was anticipated by Madison et al. mistakenly assumed that only BWM polarization angles (2014) and holds true even with the more sophisticated between 0 and π/2 must be considered. We here correctly noise modeling we have employed in this analysis. carry out a search that tests polarization angles between 10. CONCLUSION 0 and π. In this paper, we have conducted a search for BWMs With the deﬁnitions for τ and τ in place, we can E P in the ﬁrst ﬁve years of NANOGrav data. We did not derive constraints on Λ(> h ) from our Earth-term and B detect any BWMs. Based on our current understanding pulsar-term analyses: of SMBHBs, the most conventional anticipated source − of bright BWMs, it is unsurprising that we did not. −ln (1 Q) Wang et al. (2015) predict that BWMs from SMBHB Λ(> h ) < , (26) − B τ mergers exceeding amplitudes of 10 14 occur at a rate i of just a few every 105 yr. However, Cordes & Jenet where Q is the probability that at least one BWM oc- (2012) conclude that because of large uncertainties in curring at rate Λ(> h ) encounters the PTA during the things such as the inspiral rate of SMBHBs, the actual B time τ and i is a placeholder for either E or P. rate of BWMs is essentially unconstrained. Also, we i In Figure 7, we show our constraints on Λ(> h ) from stress that memory is a very general feature of bright B our Earth- and pulsar-term analyses. We have set Q = GW events and large-amplitude BWMs may be produced 0.95. We have set n = 3.52 for both our Earth- and by wholly unexpected phenomena occurring at uncon- pulsar-term constraints. This number comes from re- strained rates; ongoing searches for BWMs are crucial quiring a false alarm probability of less than ﬁve-percent for utilizing the raw discovery potential of PTA observa- ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: has

text: little

text: to

text: do

text: with

text: but

text: is

text: instead

text: because

text: these

text: sky

text: positions

text: are

text: antipodal

text: to

text: our

text: great-

text: est

text: concentration

text: of

text: pulsars,

text: especially

text: our

text: most

text: sensitive

text: pulsar,

text: The

text: green

text: triangle

text: in

text: Figure

text: in-

text: dicates

text: the

text: position

text: of

text: the

text: Virgo

text: Cluster.

text: Just

text: Mpc

text: from

text: Earth,

text: the

text: Virgo

text: Cluster

text: falls

text: very

text: near

text: the

text: edge

text: of

text: the

text: volume

text: over

text: which

text: we

text: are

text: sensitive

text: to

text: BWMs

text: from

text: I

text: µ

text: binary

text: mergers.

text: BWM

text: RATE-AMPLITUDE

text: CONSTRAINTS

text: To

text: synthesize

text: the

text: results

text: from

text: both

text: our

text: Earth-

text: and

text: pulsar-term

text: analyses,

text: we

text: derive

text: constraints

text: on

text: Λ(>

text: h

text: B

text: the

text: annual

text: rate

text: of

text: BWMs

text: from

text: any

text: part

text: of

text: the

text: sky

text: with

text: any

text: polarization

text: having

text: amplitudes

text: greater

text: than

text: h

text: as-

text: B

text: suming

text: they

text: occur

text: as

text: a

text: Poisson

text: process.

text: This

text: quantity

text: is

text: readily

text: relatable

text: to

text: astrophysical

text: models

text: of

text: processes

text: producing

text: BWMs,

text: i.e.,

text: Equation

text: of

text: Cordes

text: Jenet

text: Equation

text: of

text: Madison

text: et

text: al.

text: or

text: Equation

text: of

text: Wang

text: et

text: al.

text: Toward

text: this

text: end,

text: deﬁne

text: the

text: quantities,

text: Fig.

text: conﬁdence

text: Earth-term

text: upper

text: bound

text: on

text: the

text: rate,

text: Λ(>

text: hB),

text: of

text: BWMs

text: occurring

text: with

text: amplitudes

text: at

text: or

text: above

text: ampli-

text: ∆t∆Ω∆ψ

text: Nt

text: NΩ

text: Nψ

text: tudes

text: hB.

text: The

text: two

text: curves

text: come

text: from

text: our

text: Earth-term

text: and

text: pulsar-

text: τ

text: (h

text: n)=

text: Θ(h

text: nσ

text: term

text: analyses.

text: The

text: pulsar-term

text: probes

text: lower-rate

text: events

text: at

text: high

text: E

text: B

text: B

text: h,ijk

text: amplitudes

text: because

text: individual

text: pulsar

text: terms

text: contain

text: causally

text: inde-

text: i

text: j

text: k

text: pendent

text: information.

text: The

text: near

text: total

text: convergence

text: of

text: the

text: Earth-

text: and

text: pulsar-term

text: curves

text: at

text: low

text: BWM

text: amplitudes

text: demonstrates

text: that

text: a

text: single

text: pulsar,

text: is

text: dominating

text: our

text: sensitivity.

text: and

text: in

text: an

text: Earth-term

text: search

text: with

text: NG

text: Our

text: estimate

text: of

text: N

text: is

text: likely

text: signiﬁcantly

text: larger

text: than

text: it

text: needs

text: to

text: be.

text: A

text: G

text: detailed

text: suite

text: of

text: simulations

text: could

text: give

text: us

text: a

text: more

text: realistic

text: ∆t

text: M

text: Nt

text: nσ

text: assessment

text: of

text: N

text: which

text: would

text: lead

text: to

text: more

text: constrain-

text: τ

text: (h

text: n)=

text: Θ

text: h

text: h,p,K

text: dψdΩ,

text: G

text: P

text: B

text: B

text: B(θ,

text: φ)

text: ing

text: upper

text: limits,

text: but

text: for

text: a

text: ﬁxed

text: allowable

text: false

text: alarm

text: K

text: i

text: probability,

text: the

text: amplitude

text: a

text: BWM

text: must

text: exceed

text: to

text: be

text: inconsistent

text: with

text: the

text: noise

text: scales

text: as

text: the

text: square

text: root

text: of

text: the

text: natural

text: logarithm

text: of

text: N

text: so

text: the

text: improvement

text: to

text: the

text: G

text: upper

text: limit

text: from

text: this

text: analysis

text: would

text: be

text: very

text: slight.

text: where

text: ∆t,

text: ∆Ω,

text: and

text: ∆ψ

text: describe

text: the

text: grid

text: spacing

text: in

text: Since

text: there

text: are

text: fewer

text: eﬀectively

text: independent

text: trials

text: in

text: a

text: the

text: Earth-term

text: search.

text: The

text: quantities

text: τ

text: and

text: τ

text: are

text: the

text: E

text: P

text: pulsar-term

text: search,

text: we

text: should

text: use

text: a

text: lower

text: value

text: of

text: n

text: for

text: a

text: total

text: time

text: that

text: the

text: PTA

text: had

text: n-σ

text: sensitivity

text: to

text: a

text: BWM

text: of

text: conﬁdence

text: upper

text: limit.

text: But,

text: by

text: treating

text: the

text: Earth-

text: amplitude

text: at

text: least

text: h

text: in

text: the

text: Earth-term

text: and

text: pulsar-term,

text: B

text: and

text: pulsar-terms

text: comparably,

text: we

text: are

text: able

text: to

text: highlight

text: respectively,

text: weighted

text: by

text: the

text: fraction

text: of

text: the

text: total

text: source-

text: an

text: important

text: fact

text: about

text: our

text: data

text: set.

text: The

text: near-total

text: location

text: and

text: polarization

text: angle

text: space

text: over

text: which

text: that

text: convergence

text: of

text: our

text: Earth-

text: and

text: pulsar-term

text: constraints

text: sensitivity

text: was

text: achieved.

text: Our

text: deﬁnitions

text: for

text: τ

text: and

text: τ

text: E

text: P

text: for

text: low

text: values

text: of

text: h

text: indicates

text: that

text: our

text: upper

text: limits

text: are

text: diﬀer

text: from

text: nearly

text: identical

text: deﬁnitions

text: in

text: Madison

text: et

text: al.

text: B

text: almost

text: entirely

text: dominated

text: by

text: a

text: single

text: pulsar

text: data

text: set:

text: by

text: an

text: overall

text: factor

text: of

text: two.

text: In

text: that

text: work,

text: it

text: was

text: This

text: was

text: anticipated

text: by

text: Madison

text: et

text: al.

text: mistakenly

text: assumed

text: that

text: only

text: BWM

text: polarization

text: angles

text: and

text: holds

text: true

text: even

text: with

text: the

text: more

text: sophisticated

text: between

text: and

text: must

text: be

text: considered.

text: We

text: here

text: correctly

text: noise

text: modeling

text: we

text: have

text: employed

text: in

text: this

text: analysis.

text: carry

text: out

text: a

text: search

text: that

text: tests

text: polarization

text: angles

text: between

text: CONCLUSION

text: and

text: π.

text: In

text: this

text: paper,

text: we

text: have

text: conducted

text: a

text: search

text: for

text: BWMs

text: With

text: the

text: deﬁnitions

text: for

text: τ

text: and

text: τ

text: in

text: place,

text: we

text: can

text: E

text: P

text: in

text: the

text: ﬁrst

text: ﬁve

text: years

text: of

text: NANOGrav

text: data.

text: We

text: did

text: not

text: derive

text: constraints

text: on

text: Λ(>

text: h

text: from

text: our

text: Earth-term

text: and

text: B

text: detect

text: any

text: BWMs.

text: Based

text: on

text: our

text: current

text: understanding

text: pulsar-term

text: analyses:

text: of

text: SMBHBs,

text: the

text: most

text: conventional

text: anticipated

text: source

text: of

text: bright

text: BWMs,

text: it

text: is

text: unsurprising

text: that

text: we

text: did

text: not.

text: −ln

text: Q)

text: Wang

text: et

text: al.

text: predict

text: that

text: BWMs

text: from

text: SMBHB

text: Λ(>

text: h

text: B

text: τ

text: mergers

text: exceeding

text: amplitudes

text: of

text: occur

text: at

text: a

text: rate

text: i

text: of

text: just

text: a

text: few

text: every

text: yr.

text: However,

text: Cordes

text: Jenet

text: where

text: Q

text: is

text: the

text: probability

text: that

text: at

text: least

text: one

text: BWM

text: oc-

text: conclude

text: that

text: because

text: of

text: large

text: uncertainties

text: in

text: curring

text: at

text: rate

text: Λ(>

text: h

text: encounters

text: the

text: PTA

text: during

text: the

text: things

text: such

text: as

text: the

text: inspiral

text: rate

text: of

text: SMBHBs,

text: the

text: actual

text: B

text: time

text: τ

text: and

text: i

text: is

text: a

text: placeholder

text: for

text: either

text: E

text: or

text: P.

text: rate

text: of

text: BWMs

text: is

text: essentially

text: unconstrained.

text: Also,

text: we

text: i

text: In

text: Figure

text: we

text: show

text: our

text: constraints

text: on

text: Λ(>

text: h

text: from

text: stress

text: that

text: memory

text: is

text: a

text: very

text: general

text: feature

text: of

text: bright

text: B

text: our

text: Earth-

text: and

text: pulsar-term

text: analyses.

text: We

text: have

text: set

text: Q

text: GW

text: events

text: and

text: large-amplitude

text: BWMs

text: may

text: be

text: produced

text: We

text: have

text: set

text: n

text: for

text: both

text: our

text: Earth-

text: and

text: by

text: wholly

text: unexpected

text: phenomena

text: occurring

text: at

text: uncon-

text: pulsar-term

text: constraints.

text: This

text: number

text: comes

text: from

text: re-

text: strained

text: rates;

text: ongoing

text: searches

text: for

text: BWMs

text: are

text: crucial

text: quiring

text: a

text: false

text: alarm

text: probability

text: of

text: less

text: than

text: ﬁve-percent

text: for

text: utilizing

text: the

text: raw

text: discovery

text: potential

text: of

text: PTA

text: observa-

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 is an expansion in approximately local operators, in a quantiﬁable sense. Moreover, the ﬂow-time dependence will cancel in the product of the Wilson coeﬃcient and its associated matrix element, to the desired order in the ﬂow time. We are free to set p = p = 0 in Eq. (35), because the 1 2 smeared Wilson coeﬃcient is independent of the external momenta and the mass to the order at which we are FIG. 6. At small ﬂow times (left-hand diagram), the smeared operators are localized relative to the spacetime separation of working [53]. Expa(cid:90)nding in the mass, we obtain the nonlocal operator. At large ﬂow-time values (right-hand diagram), however, the smearing radius probes the scale of the · − − 1 eik x(cid:20) e 2k2τ (cid:18) (cid:19)(cid:21) nonlocal and the sOPE is a poor approximation to the original d(1) = nonlocal operator. For details of the Feynman diagrams, see ρ2 2 (k2)2 k the caption of Fig. 2. − 1 − x2 = γ 1 + log . (38) (cid:90) (cid:2) (cid:3) 32π2 E 8τ · − − Combining this with the leading-order contribution, d d(1) = qi eik x e (k2+q2)τ(1 + (q2 + m2)τ) , which is just unity(cid:20), we have (cid:18) (cid:19)(cid:21) dp ρ2 (k2 + m2)(q2 + m2)2 i k (36) λ − x2 O we obtain a convergent integral. The x → 0 limit of dρ2 = 1 + 32π2 γ E 1 + log 8τ + (λ2). (39) this integral is now well deﬁned and only vanishes if the ﬂow time is related to the spacetime separation. An anal- In contrast, the Wilson coeﬃcient for the OPE in the ogous result holds if we take a derivative with respect to MS scheme, den(cid:20)oted by c, is (cid:18) (cid:19)(cid:21) the external mass, m. On dimensional grounds, then, we guarantee that the smeared Wilson coeﬃcient is indepen- ∝ λ µ2x2 O dent of the external states by choosing τ x2. c = 1 + 1 + 2γ + log + (λ2). (40) On physical grounds, however, we require that the φ2 32π2 E 16 smearing radius is smaller than the spacetime extent of the nonlocal operator: s < x. This choice ensures that We note the occurrence of the three features we ob- rms the sOPE remains an expansion in local operators. Phys- served for the leading-order disconnected contribution: ically speaking, if the gradient ﬂow probes length scales the same spacetime dependence for both smeared and un- on the order of the nonlocal operator, which we represent smeared coeﬃcients; the appearance of the ﬂow time as in Fig. 6, then smeared operators would cease to be (ap- a leading-order regulator; and the need to choose a small proximately) local. In other words, the sOPE would be ﬂow time for small spacetime separations to avoid large a poor expansion for the original operator. This is the logarithmic contributions. From the small ﬂow-time ex- physical origin of the third feature that we saw in the pansion viewpoint, we can conﬁrm that the ﬂow-time de- leading-order disconnected contribution, Eq. (25): small pendence ultimately cancels to the desired order. For spacetime separations require small ﬂow times to ensure example, the matrix element of ρ2(τ, 0) coupled to two convergence. The OPE requires small spacetime separa- ex(cid:10)ternal ﬁelds is (cid:11) (cid:101) (cid:101) tions for good convergence, so it follows that the sOPE requires small ﬂow times as well. | (cid:2) | (cid:0) (cid:1)(cid:3) There is a complementary viewpoint that elucidates Ω ρ2(τ, 0)φ(p1)φ(p2) Ω = 1 the small ﬂow-time requirement more quantitatively: the λ O + 1 + γ + log 2m2τ + (τ, λ2) (41) small ﬂow-time expansion [31, 50, 52]. In this case we 32π2 E see that the derivative of Eq. (36) is independent of the → external momenta in the x 0 limit, up to terms linear for suﬃciently small ﬂow times. Here we have dropped in the ﬂow time(cid:90): (cid:0) (cid:1) the external ﬁelds, which we take to be on shell, for sim- plicity. The product of this matrix element with the Wil- · − ddpi d(ρ12) = k (k2 +qime2ik)(xq2 +1m2)2 + O(τ). (37) saonndcOo(cid:10)e(ﬃτ)c,ieanstwise(cid:101)iwndouepl(cid:101)denedxepnetc(cid:11)to:f the ﬂow time to one loop (cid:20) (cid:18) (cid:19)(cid:21) | | From this, it is clear that the Wilson coeﬃcients will d Ω ρ2(τ, 0)φ(p )φ(p ) Ω = 1 ρ2 1 2 be independent of the external states, up to terms lin- λ m2x2 O ear in the ﬂow time. Therefore, provided the ﬂow time + 2γ + log + (τ, λ2). (42) is small relative to the spacetime separation, the sOPE 32π2 E 4 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: an

text: expansion

text: in

text: approximately

text: local

text: operators,

text: in

text: a

text: quantiﬁable

text: sense.

text: Moreover,

text: the

text: ﬂow-time

text: dependence

text: will

text: cancel

text: in

text: the

text: product

text: of

text: the

text: Wilson

text: coeﬃcient

text: and

text: its

text: associated

text: matrix

text: element,

text: to

text: the

text: desired

text: order

text: in

text: the

text: ﬂow

text: time.

text: We

text: are

text: free

text: to

text: set

text: p

text: p

text: in

text: Eq.

text: because

text: the

text: smeared

text: Wilson

text: coeﬃcient

text: is

text: independent

text: of

text: the

text: external

text: momenta

text: and

text: the

text: mass

text: to

text: the

text: order

text: at

text: which

text: we

text: are

text: FIG.

text: At

label: caption

text: small

text: ﬂow

text: times

text: (left-hand

text: diagram),

text: the

text: smeared

text: operators

text: are

text: localized

text: relative

text: to

text: the

text: spacetime

text: separation

text: of

text: working

label: equation

text: in

text: the

text: mass,

text: we

text: obtain

text: the

text: nonlocal

text: operator.

text: At

text: large

text: ﬂow-time

text: values

text: (right-hand

text: diagram),

text: however,

text: the

text: smearing

text: radius

text: probes

text: the

text: scale

text: of

text: the

text: eik

text: e

text: nonlocal

text: and

text: the

text: sOPE

text: is

text: a

text: poor

text: approximation

text: to

text: the

text: original

text: nonlocal

text: operator.

text: For

text: details

text: of

text: the

text: Feynman

text: diagrams,

text: see

text: ρ2

text: k

text: the

text: caption

text: of

text: Fig.

text: γ

text: log

text: E

text: 8τ

text: Combining

text: this

text: with

text: the

text: leading-order

text: contribution,

text: d

text: qi

text: eik

text: x

text: e

text: which

text: is

text: just

text: we

text: have

text: dp

text: ρ2

text: i

text: k

text: λ

text: O

text: we

text: obtain

text: a

text: convergent

text: integral.

text: The

text: x

text: limit

text: of

text: γ

text: E

text: log

text: 8τ

text: this

text: integral

text: is

text: now

text: well

text: deﬁned

text: and

text: only

text: vanishes

text: if

text: the

text: ﬂow

text: time

text: is

text: related

text: to

text: the

text: spacetime

text: separation.

text: An

text: anal-

text: In

text: contrast,

text: the

text: Wilson

text: coeﬃcient

text: for

text: the

text: OPE

text: in

text: the

text: ogous

text: result

text: holds

text: if

text: we

text: take

text: a

text: derivative

text: with

text: respect

text: to

text: MS

text: scheme,

text: by

text: c,

text: is

text: the

text: external

text: mass,

text: m.

text: On

text: dimensional

text: grounds,

text: then,

text: we

text: guarantee

text: that

text: the

text: smeared

text: Wilson

text: coeﬃcient

text: is

text: indepen-

text: λ

text: O

text: dent

text: of

text: the

text: external

text: states

text: by

text: choosing

text: τ

text: c

text: 2γ

text: log

text: On

text: physical

text: grounds,

text: however,

text: we

text: require

text: that

text: the

text: φ2

text: E

text: smearing

text: radius

text: is

text: smaller

text: than

text: the

text: spacetime

text: extent

text: of

text: the

text: nonlocal

text: operator:

text: s

text: x.

text: This

text: choice

text: ensures

text: that

text: We

text: note

text: the

text: occurrence

text: of

text: the

text: three

text: features

text: we

text: ob-

text: rms

text: the

text: sOPE

text: remains

text: an

text: expansion

text: in

text: local

text: operators.

text: Phys-

text: served

text: for

text: the

text: leading-order

text: disconnected

text: contribution:

text: ically

text: speaking,

text: if

text: the

text: gradient

text: ﬂow

text: probes

text: length

text: scales

text: the

text: same

text: spacetime

text: dependence

text: for

text: both

text: smeared

text: and

text: un-

text: on

text: the

text: order

text: of

text: the

text: nonlocal

text: operator,

text: which

text: we

text: represent

text: smeared

text: coeﬃcients;

text: the

text: appearance

text: of

text: the

text: ﬂow

text: time

text: as

text: in

text: Fig.

text: then

text: smeared

text: operators

text: would

text: cease

text: to

text: be

text: (ap-

text: a

text: leading-order

text: regulator;

text: and

text: the

text: need

text: to

text: choose

text: a

text: small

text: proximately)

text: local.

text: In

text: other

text: words,

text: the

text: sOPE

text: would

text: be

text: ﬂow

text: time

text: for

text: small

text: spacetime

text: separations

text: to

text: avoid

text: large

text: a

text: poor

text: expansion

text: for

text: the

text: original

text: operator.

text: This

text: is

text: the

text: logarithmic

text: contributions.

text: From

text: the

text: small

text: ﬂow-time

text: ex-

text: physical

text: origin

text: of

text: the

text: third

text: feature

text: that

text: we

text: saw

text: in

text: the

text: pansion

text: viewpoint,

text: we

text: can

text: conﬁrm

text: that

text: the

text: ﬂow-time

text: de-

text: leading-order

text: disconnected

text: contribution,

text: Eq.

text: small

text: pendence

text: ultimately

text: cancels

text: to

text: the

text: desired

text: order.

text: For

text: spacetime

text: separations

text: require

text: small

text: ﬂow

text: times

text: to

text: ensure

text: example,

text: the

text: matrix

text: element

text: of

text: coupled

text: to

text: two

text: convergence.

text: The

text: OPE

text: requires

text: small

text: spacetime

text: separa-

text: ﬁelds

text: is

text: tions

text: for

text: good

text: convergence,

text: so

text: it

text: follows

text: that

text: the

text: sOPE

text: requires

text: small

text: ﬂow

text: times

text: as

text: well.

text: There

text: is

text: a

text: complementary

text: viewpoint

text: that

text: elucidates

text: Ω

text: Ω

text: the

text: small

text: ﬂow-time

text: requirement

text: more

text: quantitatively:

text: the

text: λ

text: O

text: γ

text: log

text: (τ,

text: small

text: ﬂow-time

text: expansion

text: In

text: this

text: case

text: we

text: E

text: see

text: that

text: the

text: derivative

text: of

text: Eq.

text: is

text: independent

text: of

text: the

text: external

text: momenta

text: in

text: the

text: x

text: limit,

text: up

text: to

text: terms

text: linear

text: for

text: suﬃciently

text: small

text: ﬂow

text: times.

text: Here

text: we

text: have

text: dropped

text: in

text: the

text: ﬂow

text: the

text: external

text: ﬁelds,

text: which

text: we

text: take

text: to

text: be

text: on

text: shell,

text: for

text: sim-

text: plicity.

text: The

text: product

text: of

text: this

text: matrix

text: element

text: with

text: the

text: Wil-

text: ddpi

text: k

text: O(τ).

text: the

text: ﬂow

text: time

text: to

text: one

text: loop

text: From

text: this,

text: it

text: is

text: clear

text: that

text: the

text: Wilson

text: coeﬃcients

text: will

text: d

text: Ω

text: )φ(p

text: Ω

text: ρ2

text: be

text: independent

text: of

text: the

text: external

text: states,

text: up

text: to

text: terms

text: lin-

text: λ

text: O

text: ear

text: in

text: the

text: ﬂow

text: time.

text: Therefore,

text: provided

text: the

text: ﬂow

text: time

text: 2γ

text: log

text: (τ,

text: is

text: small

text: relative

text: to

text: the

text: spacetime

text: separation,

text: the

text: sOPE

text: E

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##