# Presentation: 802347

## Theoretical Predictions of GZK Neutrinos:An Experimentalist’s Perspective

**Amy Connolly**
**11/07/03**

## Outline

- (Briefly) What is a GZK neutrino
- Good papers to read
- How to Calculate a GZK  flux
- The parameters of the models
- Predicted fluxes
- Max/min scenarios
- Experimental predictions
- Conclusions

## Ultra-high energy protons would interact with CMB photons
Photoproduction:
p + CMB  →  n +  →  p e-  e   e+ e  
 			     →  p 0  →  p  
Threshold:  p > 3 × 1019 eV
Delta resonance:
p + CMB → (1232) → n + → p e-  e   e+ e  
 				         →  p 0 → p 
Threshold:  p > 6.8 × 1020 eV
Numbers to that are good to know:
50 Mpc = 163 Mly,  z = 0.012
 n(En=1020 eV) = 3My = 1Mpc
 n(E=1018 eV) = 10-30 cm2

- Photoproduction:
- p + CMB  →  n +  →  p e-  e   e+ e  
-  			     →  p 0  →  p 
- Threshold:  p > 3 × 1019 eV
- Delta resonance:
- p + CMB → (1232) → n + → p e-  e   e+ e  
-  				         →  p 0 → p 
- Threshold:  p > 6.8 × 1020 eV
- Numbers to that are good to know:
- 50 Mpc = 163 Mly,  z = 0.012
-  n(En=1020 eV) = 3My = 1Mpc
-  n(E=1018 eV) = 10-30 cm2

- What is a GZK Neutrino

-  Protons from Source X
-  would not interact before arrival,
- Protons from Source Y
- would

## Good Papers to Read/Papers Referred to in This Talk

- R. Engel, D. Seckel, T. Stanev, ``Neutrinos from Propagation of Ultrahigh Energy Protons,’’  Phys.Rev. D64:093010
- Z. Fodor, S.D.Katz, A.Ringwald, and H.Tu, ``Bounds on the Cosmogenic Neutrino Flux,’’ hep-ph/0309171
- David’s documentation on UHE neutrino telescopes and reporting results
- J.Bahcall, E.Waxman, ``High Energy Astrophysical Neutrinos:  the Upper Bound is Robust,’’ hep-ph/9902383
- E.Waxman, Astrophys.J.Lett.452,L1(1995)
- C.T.Hill and D.N.Schramm, ``The Ultrahigh-Energy Cosmic Ray  Spectrum,’’ Phys.Rev.D31,564(1985)

## How to Calculate a GZK  flux

- Hypothesize proton sources at cosmological distances
-  proton luminosity L (E, z) = J(E) × (z)
-  J(E):  injected power  [E/T/V]
-  (z):  cosmological evolution (dimensionless)
-  Emax:  maximum production energy
-  zmin, zmax:  minimum, maximum source redshift
- Pick a universe: cosmological expansion (z)=dt/dz  [T]
- Propogate the protons to the earth
-  get number of neutrinos per proton per bin in E
- Y(Ep,En,z):  yield function [#/#p/E]
- Integrate over Ep, z to get the neutrino *brightness*

- [#/E/A/T/sr]

## Cosmological evolution parameterizes our ignorance about the nature of the sources
Usually takes the form:
 r(z)=(1+z)n (z-zmin) (zmax-z)           ( is step function)
Typically, n≈3, zmin=0.012, zmax≈2 are chosen
Cosmological evolution parameters contain the answers to
How  young was the universe when the sources began to emit UHE protons (zmax)?
Do younger sources behave differently than older ones, How are sources radially distributed (n)?
How close to Earth are the closest sources (zmin)?
Motivation for power law (or n) difficult to find
(Hill, Schramm, 1985):  power law characteristic of ``bright- phase’’ models
Example:  observed quasi-stellar objects at z=2 with 100× activity compared to z=0 → power law reasonable
 zmax also not well motivated, zmin=0.012 reasonable (50 Mpc)
Increase n or zmax→ increase flux

- Usually takes the form:
-  r(z)=(1+z)n (z-zmin) (zmax-z)           ( is step function)
- Typically, n≈3, zmin=0.012, zmax≈2 are chosen
- Cosmological evolution parameters contain the answers to
- How  young was the universe when the sources began to emit UHE protons (zmax)?
- Do younger sources behave differently than older ones, How are sources radially distributed (n)?
- How close to Earth are the closest sources (zmin)?
- Motivation for power law (or n) difficult to find
- (Hill, Schramm, 1985):  power law characteristic of ``bright- phase’’ models
- Example:  observed quasi-stellar objects at z=2 with 100× activity compared to z=0 → power law reasonable
-  zmax also not well motivated, zmin=0.012 reasonable (50 Mpc)
- Increase n or zmax→ increase flux

- Cosmological Evolution of the Proton Sources (z), zmin, zmax

## Cosmological Expansion (z)

-  Contains the cosmological parameters
- Recall that
- T→ T/(1+z), L→L/(1+z)
- For flat universe (M+=1):   (z) cannot deviate from Einstein-de Sitter by more than factor
- (ES&S):  To claim that  increases flux may be too optimistic
- Keeping (z)×(z) constant may be more realistic

## Propagating the Protons to Earth

- Commonly a MC program is used (SOPHIA)
- Photoproduction interaction model
- Fits well with multiparticle production data over wide energy range
- UHE protons interact quickly!
- Minimum mean free path at     ~6×1020 GeV → 3.8 Mpc
- ~40% of energy goes to neutrinos Ep>1021 eV
- For neutrons < 4×1020 eV:
-  interaction length > c
-  lower energy neutrons decay to e
- FKR&T have put propagation functions P|p(E;Ep,r) at _[www.desy.de/~uhecr](http://www.desy.de/~uhecr)_
- Probability proton created at r with energy Ep gives a neutrino with E>E

- (1232)
- threshold

- E/Ep

- Proton Energy (eV)

- 10 to
- 200
- Mpc

## Injected Power J(E)

- Energy ejected from the source (as protons) per volume per time
- J(E) = J0 × f(E), usually f(E) ~ E-
- Increase → flux shifts low
- J0:  Normalization
- Rough estimate (outlined by Engel, Seckel, Stanev, 2001)
- Take cosmic ray flux at 1019 eV (2.5 × 10-28 /cm2/s/sr/GeV)
- Assume all extragalactic at that energy
- Assume  =2
- Lifetime = Hubble time (CR=1010 yr)
- Find J0=1.1 × 1044 erg/Mpc3/yr per decade of energy
- (Waxman, 1995) – often referenced:
- Fit , J0 such that after propagation, gives observed CR flux
- Find = 1.8-2.7 and
- J0 = 4.5 ± 1.5 × 1044 erg/Mpc3/yr as 1019-1021 eV protons
- Based on old data sample – only 2 events >1020 eV

- ES&S use
- Waxman flux for
- their calculations

## Injected Power J(E) - cont

- J0 × f(E):
- ES&S:
- Take J0 from Waxman and =2
- (Fodor, Katz, Ringwald, Tu, 2003):
- Events > 1020 eV cannot be fit to power law injection spectrum
- Method 1:
- Fixed J0 (independent of ) from observed CR flux
- Fit , n, Emax to observed CR spectrum only to E+=20 GeV
- AGASA:  Emax=3 × 1021 eV, =2.57, n=3.30
- Fly’s Eye: Emax=3 × 1021 eV, =2.50, n=3.80
- Method 2:  Consider
- entire spectrum (maximum),  only “pileup” (minimum)
- invert propagation functions to get injected power
-  parameter: n
- No J0,  to speak of

## Predicted Fluxes

- ES&S:
- J0= 4.5 ± 1.5 × 1044 erg/Mpc3/yr (Waxman)
-  =2, zmin=0, zmax≈2.7, n=3 (flat in 1.9<z<2.7)
-  M=1, =0
- SOPHIA for propagation

-  log E2 dN/dE/cm2/s/sr

-  log E dN/dE/cm2/s/sr

- Neutrino Energy (eV)

- ES&S

- ES&S

- Waxman & Bahcall:  Similar parameters EXCEPT assume entire proton energy goes to neutrinos

- FKR&T:
- Best fits to data
- Emax=3×1021eV
-  zmin=0.012, zmax=2

- =2.57, n=3.30

- =2.50, n=3.80

- [eV/m2/s/sr]

- [1/cm2/s/sr]

- ×10-4/E to get
-  the same units as ES&S

- FKR&T

- FKR&T

## Defining Max/Min Scenarios

- Criteria for min/max scenarios non-trivial
- Flux increases if you increase:
-  n, zmax, or (maybe)
-  number of cosmogenic protons >1020 eV
-   shifts flux to lower energy
-    increased 2→3, <E> →<E>/10
- Possible minimal scenarios:
- 1/2 CR sources local (a scenario in ES&S)
- Only neutrinos in the “pile-up” are cosmogenic (FKR&T)

- (green)  n=4
- (black) n=3

- FKR&T

-  log E2 dN/dE/cm2/s/sr

- [eV/m2/s/sr]

- [1/cm2/s/sr]

- Neutrino Energy (eV)

-  log E2 dN/dE/cm2/s/sr

- Neutrino Energy (eV)

- log E dN/dE/cm2/s/sr

- Can we all agree
-  on the same units??

- log E dN/dE/cm2/s

- n=4
- zmax=8
- =0.7

- Flux decreased by factor 2

- Integrated over 4!!

## Expected Events from Experiments

- David has produced [A]t/N90 plots
- Simply fold in flux from theory
- ANITA (45 days livetime):
- Assume same sensitivity for all 3 flavors
- ES&S flux:  3 (4.8)** events
- ES&S maximal:  18 events
- n=4, =2, zmax=8, =0.7
- ES&S minimal:  1.5 (2.4) events
- FDKR&T (best fit):  2 events
- FDKR&T (minimal):  0.3 events
- FDKR&T (maximal):  150 events!
- ** discrepancy between my calculation and the ANITA proposal still being worked out.

- Log(Aefft/km2-sr-s)

-  log(E/eV)

## Conclusions

- Many parameters go into GZK calculations
- Those that have the largest effect are:
-  cosmological evolution (n, zmax)
-  injected power spectrum index 
-  number of observed CR protons assumed to be cosmogenic
- Reasonable criteria for min/max scenarios difficult
-  hard lower and upper bounds seem extreme
- ANITA expected events range from ~few to          ~150 (extreme case)
- Neutrino experiments may tell us some interesting information about the sources of CR’s
-  n, zmax,  all contain interesting physics