Outline 1) Incorporating theoretical systematics in p fits 2) - - PowerPoint PPT Presentation

Outline

Is DM hidden in the ¯ p flux?

Pierre

1) Incorporating theoretical systematics in ¯ p fits 2) Spectral anomalies – Frequentist vs Bayesian 3) A few recent examples

CRAC meeting – Grenoble – July 5, 2019

CR space ce CR + DM space

Background ⌘ no DM DM

χ2

bckg

χ2

¯ p

and χ2

DM

In general systematics ≡ CR + {Φi = Φp + Φα} + dσij→¯

p

dE¯

p

1) Incorporating theoretical systematics in ¯ p fits

Background ⌘ no DM DM

χ2

¯ p

χ2

bckg

and χ2

DM

global space CR space ce CR + DM space ce global + DM space

In general systematics ≡ CR + {Φi = Φp + Φα} + dσij→¯

p

dE¯

p

1) Incorporating theoretical systematics in ¯ p fits

Incorporating CR systematics in the ¯ p fit

Background only vs DM hypotheses

(i) Old days agnostic scan of CR parameter space to find χ2

¯ p , min

P(¯ p | flat θCR) ⇒ loss of information from B/C (ii) Fit together ¯ p and B/C to get best-fit CR parameters θCR P(¯ p + B/C | θCR) ⇒ not the same question Fit could be dominated by B/C and not ¯ p (iii) Fit ¯ p and incorporate a theoretical CR uncertainty matrix such as C¯

p source nm

= D Φth

¯ p

• n
• Φth

¯ p

• m

E − ⌦ Φth

¯ p

↵

n

⌦ Φth

¯ p

↵

m with CB/C matrix

C¯

p nm = CAMS nm

+ C¯

p source nm

(iv) Use θCR as nuisance parameters when fitting ¯ p data χ2 = χ2

¯ p +

• χ2

CR ≡ (θp − ¯

θp) CCR

pq (θq − ¯

θq) How is CCR

pq defined? If CCR ≡ CB/C ⇒ back to (iii)

Bayesian with P(θCR | ¯ p) = L(¯ p | θCR) × Π(θCR)

2) Spectral anomalies – Frequentist vs Bayesian

(a) Model Mi is wright or wrong depending on its reduced χ2 per d.o.f. Frequentist approach – models M1 and M2 are both true Maximum likelihood method if M1 ⊂ M2 (b) Generate Mock data sets with best-fit model M1, fit them and look for the Monte Carlo distribution of ∆χ2 = χ2

1 − χ2 2 .

Conclude from P(∆χ2 | M1) Mock data to be generated – lengthy procedure Frequentist approach – may be the best one (c) Use the Akaike information criterion (AIC) and compute the difference ∆AIC = AIC2 − AIC1 . AIC = 2 p − 2 ln Lmax with p parameters AICcor = AIC + 2 p2 + 2 p N − p − 1 with N data points In our example, N is large and ∆AIC = 4 − ∆χ2 P(M1) P(M2) = e∆AIC/2 In Cuoco et al. (2019) P(no DM)/P(DM) = 1.3 × 10−2

(d) Bayesian approach in the spirit of method iv with nuisance parameters. In this example θ1 = {θCR} while θ2 = {θCR, hσannvi, mχ}. Compute the probability P(¯ p | Mi) and use Bayes factor K. P(θ1 | ¯ p, M1) = L( ¯ p | θ1, M1) Π(θ1|M1) P(¯ p | M1) P(¯ p | M1) = Z L( ¯ p | θ1, M1) Π(θ1|M1) dθ1 P(M2 | ¯ p) P(M1 | ¯ p) = ⇢ K ⌘ P(¯ p | M2) P(¯ p | M1) ⇢P(M2) P(M1) ' 1

• (e) Use the Schwarz/Bayes information criterion (SBIC) and compute the

difference ∆SBIC = SBIC2 SBIC1 . SBIC = p ln(N) 2 ln Lmax with p parameters In our example, N = 158 and ∆SBIC = 10.1 ∆χ2 P(M1 | ¯ p) P(M2 | ¯ p) = e∆SBIC/2 In Cuoco et al. (2019) P(no DM)/P(DM) = 0.28 (no comment)

Jeffreys’ scale

(d) Bayesian approach in the spirit of method iv with nuisance parameters. In this example θ1 = {θCR} while θ2 = {θCR, hσannvi, mχ}. Compute the probability P(¯ p | Mi) and use Bayes factor K. P(θ1 | ¯ p, M1) = L( ¯ p | θ1, M1) Π(θ1|M1) P(¯ p | M1) P(¯ p | M1) = Z L( ¯ p | θ1, M1) Π(θ1|M1) dθ1 P(M2 | ¯ p) P(M1 | ¯ p) = ⇢ K ⌘ P(¯ p | M2) P(¯ p | M1) ⇢P(M2) P(M1) ' 1

• (e) Use the Schwarz/Bayes information criterion (SBIC) and compute the

difference ∆SBIC = SBIC2 SBIC1 . SBIC = p ln(N) 2 ln Lmax with p parameters In our example, N = 158 and ∆SBIC = 10.1 ∆χ2 P(M1 | ¯ p) P(M2 | ¯ p) = e∆SBIC/2 In Cuoco et al. (2019) P(no DM)/P(DM) = 0.28 (no comment)

3) A few recent examples

Method (ii) used to look for DM with P(p + α + ¯ p/p | θCR)

• A. Cuoco et al, Phys. Rev. D99 (2019) 103014

3) A few recent examples

Method (ii) used to look for DM with P(p + α + ¯ p/p | θCR)

• A. Cuoco et al, Phys. Rev. D99 (2019) 103014

3) A few recent examples

Method (ii) used to look for DM with P(p + α + ¯ p/p | θCR)

• A. Cuoco et al, Phys. Rev. D99 (2019) 103014

P(no DM | data) P(DM | data) = e∆SBIC/2 = 0.28

⇒

Method (e), i.e., Schwarz/Bayes information criterion, yields

• A. Reinert & M.W. Winkler, JCAP 1801 (2018) 055

Mixed methods (ii) + (iii) used to look for DM

Uncertainties on XS treated with Σ¯

p source nm

+ ΣB/C source

nm

P(¯ p + B/C + AMS/PAMELA | θCR + φF

¯ p)

Method (b) used Mock data ⇒ ∆χ2 & P(∆χ2 | M1)

⇒

Method (e), i.e., Schwarz/Bayes information criterion, yields ∆SBIC = 2 ln(N) − 2∆ln Lmax = 9.9 − ∆χ2

DM is 7.4% (¯ bb) and 2.4% (WW) as probable as no DM!

P(no DM | data) P(DM | data) = e∆SBIC/2 = 13.5 & 42.5