- Email: [email protected]
Low frequency gravity wave spectra generated by cosmic strings
Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
LOW FREQUENCY GRAVITY WAVE SPECTRA GENERATED BY COSMIC STRINGS Roger W. ROMANI Depart,wnt of Astronomy, University c~fCali/brnia, Berkeley, CA 94720, USA Received 2 June 1988
Gravity wave spectra generated by populations of cosmic string loops are computed, showing that loop evolution and radiation properties have important effects at observable wavelengths. The spectra are compared with limits derived from various observations; bounds from millisecond pulsar timing are becoming severe for several string scenarios, but implied constraints on string parameters depend sensitively on details of the loop population and dynamics.
1. Introduction
An important aspect of scenarios which seed galaxy formation and produce large scale structure from the interactions of primordial cosmic strings lies in their testable predictions for an attendant stochastic cosmological background of gravitational waves [ 1 ]. At present the most sensitive constraints on this spectrum come from pulsar timing (at wave periods 1-104 yr) and from cosmological isotropy (periods ~ 10 ~° yr). Previous work has described the qualitative features o f the gravity wave spectrum for a population of idealized non-evolving strings [2], and calculated the total energy deposited in gravity waves during the radiation era for more refined models [ 3 ]. However, because of structure imprinted on the spectrum during the transition from the radiation era to matter domination, somewhat more realistic models of string evolution may have substantially modified spectra in observationally accessible regions. In this letter, we compute the expected stochastic gravitational wave backgrounds resulting from populations of cosmic strings radiating and evolving according to several string modes, considering a range of parameters that should delineate the expected behaviour. We find that there are indeed substantial variations in the shape of the spectrum for periods > 1 yr, and since present observational bounds are quite close to the predicted fluxes, such modifications are important for comparison with specific string evolution scenarios. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
2. Gravity waves from loop populations We consider a population of long-lived (i.e. nonself-intersecting) loops following the cosmologically prescribed scaling behaviour which have a volume rate o f formation dni =Zt7 4 dti,
(I)
where a closed loop formed at time t, has a characteristic radius ri=c~ti, a typical total length fir,, and a mass per unit l e n g t h / t = e/G. These loops can decay through a variety of radiation processes; however, insofar as loops of interest are invariably massive, substantial amounts of energy will be deposited in gravitational waves. In computing the stochastic gravitational wave background one must, in general, consider the power spectrum o f the radiation emitted by each loop trajectory, average over all trajectories, average over an initial spectrum of loop sizes, follow the decay of the loops as they radiate and compute the evolution of the radiation spectrum to the present epoch. In the present context we are particularly interested in the shape of the resultant spectrum at low ( < y r - ~) frequencies, where evolution o f the loops and the cosmological expansion will be most important. The values of geometrical parameters, such as c~, and indeed the presence of a scaling solution, can be established by numerical simulations [4,5 ] and we will borrow values suggested by these studies. The gravitational radiation spectrum of an individual loop and evolution o f the loop trajectories also depend on 477
Volume 215, number 3
PHYSICS LETTERS B
details of the particle physics. However, the qualitative behaviour of loop decay can be simply written for a number of theories and we consider several models to display a range of forms for the final spectra. Several types of string are commonly discussed. In the standard picture emission of gravitational radiation causes string decay with an energy loss rate d E / d t = YGwG~ 2, where numerical work has shown that 7Gw ~ 50 [ 3 ]. In addition, strings can be "superconducting" and, if they thread sufficient flux, electromagnetic radiation losses from the persistent currents can be substantial. The magnitude of this emission appears rather more sensitive to the details of the loop trajectories than that of the gravitational loss channel. In particular, if discontinuities such as cusps or kinks are dominant sites of radiation loss, then the electromagnetic emission will be a roughly constant additional loss factor. The resultant spectrum will simply be that of a collection of loops with an increased ?Gw, renormalized to account for the lower gravitational radiation efficiency. However, if the electromagnetic losses are dominated by the low order modes of the loop, then the emission rate will be determined by the threaded magnetic field which will increase as the loop shrinks. This leads to a varying total radiation rate d E / d t = 7 ~ w [ 1 + F o ( r J r ) 2] , Fo = ( Io/ Icrit ) 2 ( YEM/ YGw )e2p2 / G lA ,
(2)
where p, ?EM/PGW~ 1 [6]. The initial current Io thus grows as the loop shrinks until a critical value Icrit is reached and it becomes energetically favorable to lose the topological twist in the field defining the string; the loop catastrophically decays into energetic particles or "quenches". To compute the gravity wave spectra resulting from these loops, we refer quantities to the era of matterradiation equality. Redshifting the initial loop density ( 1 ) to an epoch of gravity wave emission at z', and using the string density and emissivity above, one obtains the gravity wave emission rate deeq/dt' = (zeq/z')47GW
G/u2(z'/zi)3dni,
(3)
where the energy density is referenced to equality; Zeqm4.2XlO4(ff2hloo)-ZT4.7, with a Hubble constant of 100hloo k m s - l M p c -1 and a microwave 478
22 December 1988
background temperature of 2.7T2. 7 K. For a given wave frequencyfeq and a given radiating mode m, the time of emission t' determines the size of the radiating loop as r' = m [ 2 (z'/zeq )f~q ] - 1. For a loop radiating and shrinking as above this size can be related to the time of the string formation through the decay law (2), since d r / d t = - ( l / f l ~ t ) d E / d t . The radiation rate (2) gives an emission time of t'=ti
1 + ~
\at,]
-o
1-F~/2tan-lFff tan-
1
1/2
r'
which implicitly determines t' (the purely gravitational decay corresponds to F0 = 0). Taking the derivatives of (4) with respect tOfeq and t' allows one to solve for dfeq/dt'. The ratio of the emitted flux (3) to this quantity gives the gravity wave spectrum at equality as a function of the time of loop formation ti. For each frequency this may be integrated over the allowed range of G where ti.min corresponds to radiation atfeq at birth and t~.... is either the time of quench or the present time, to (for strings still radiating). The resulting gravity wave spectrum may be expressed as a fraction of the microwave background energy density ey= 3g/32Gt2q. Since the gravity waves redshift with the photons, ~2cw/~2v= (d6/dJ) eofeq/~v,eqand the present day spectrum as a function of gravity wave period is given by redshifting the wavelengths to to. Numerical simulations of evolving populations of loops in ref. [4] suggest Z~ 1, a ~ 0 . 1 and r ~ 10. Adopting these values as illustrative, we follow the prescription above to compute gravity wave the spectrum of a population of evolving strings. Gravity wave emission rates have also been computed for certain exact loop trajectories [ 3 ], for which in higher modes scales roughly as P,n ~ m-4/3. The relevance of these estimates for more general loop trajectories is uncertain; kinks may increase the loop emissivity while cosmological expansion and radiation back-reaction may serve to decrease the power emitted at higher frequencies. For the total wave power we use YGw-- 50; a similar total emission rate has been found for loops with kinks [ 7 ]. In fig. 1 we display spectra computed from the values with G/2=10 -6, 10 -s and F o = 0 (purely gravitational decay), Fo= 1.0 (fairly strong electromagnetic decay, with initial current
Volume 215, number 3
PHYSICS LETTERS B
'
'
'
I
'
'
'
I
'
'
'
p
.-""""" -2
I
'
22 December 1988
'
'
I
'
'
'
I
'
'
"' " " -
-4/a
V
BR\
PSR
........
* *
c
."" ~
,Gu = 10 -n
-3
"\
-;/
-4
- .........................
-a
J
.
.
I
.
.
i
I
I
2
. .""
I
i
I
I
i
0
I
/2~
I
_8
10-8
I
2
z
i
I
4
I
I
i
I
6
I
8
Log(P (yr))
Fig. 1. Gravitational radiation backgrounds generated by cosmic strings undergoing gravitational decay (G,u= 10 -6, 10-s; full lines) and "superconducting" decay (Fo= l; G,u= 10-6, 10-8; dashed lines). With a Pm~m 4/3 mode spectrum for the loop emission, the G/~= l0 -6 curves are modified as shown (dotted lines). Also displayed are bounds from pulsar timing and l0 ° background isotropy. 3 × 10-3Icrit). To display the change in spectral slopes for loops reaching toq these solid line curves show spectra for loops radiating in the fundamental only, in addition for the G/Z= 10 -6 strings we show the hardening of the spectra resulting from a loop spectrum P,, ~ m -4/3 (dashed lines). The qualitative features of such spectra have been described in refs. [ 1,2 ]. At high frequencies the most important strings are born and decay entirely in the radiation dominated epoch, thus channeling a fixed fraction of their energy density into gravity waves and giving a flat ~ p o spectrum. For 7GwG/Z<< 1, Fo>~ 1 and m = 1, eq. (2) may be approximated as t' ~ { 1 [2(z'/Zeq)~q]-3}tio'fl/3Fo~GwG/z. So the loop decay accelerates at t'*~tic~fl/3FoTGwG/z and the dominant contribution to the spectrum at f~q comes from loops formed at ~ 4o~2)c eq 2 t e q -/ I- , , . With these estimates, we can approximate the derivative d f ~ q / d t and the integral f (decq/dt') (dt'/dfcq ) dti, yielding
-Qe~w~t@a-zr(o/-fl)3/2(G/z/TGW)l/2(3Fo)-3/2g2v for feq >>
(fl/3FoaT~wG/z)l/2/2teq.
(5)
When Fo < 1,
(3Fo) - 3/2 is replaced by unity and eq. (5) agrees with the results of ref. [3]. At very low frequencies, the spectra follow £2GWOC(G/Z)2(p/to)-1 and the dominant loops were born in the matter era. However, at intermediate frequencies, the important strings are those born in the radiation era and surviving into the period of matter domination; accordingly the spectra in this region depend on the evolution and radiation properties o f the loops and simple estimates based on the loop lifetime and the total loop energy density may not be sufficiently accurate. Since 102 yr~> teqZeq/G/z , the waves in this region can be of interest to pulsar timing experiments. As the details of the string evolution are rather poorly known, constraints in this spectral region should properly be compared to specific scenarios. Conversely, if waves in this region should be detected, they might then provide some means of discriminating between various modes of loop decay. The importance of the uncertainties in the basic parameters is depicted in fig. 2, where we reproduce the purely gravitational decay, G/z= 10 .6 curve of fig. 479
Volume 215, number 3
PHYSICS LETTERS B
-
22 December 1988
,."" .....
.,'"
PSR
-
""-
a=SxlO-a
P m ~ m -2 ~
"'-,
\CBR
\
-1 __"
*PSR),". , ' .... ;
....
,
..............
-2 v O -1
t
-3
u
I
t
Fo=O.I
-4
-15 -2
I
I
,
I o
,
,
,
I 2
,
,
,
Log(P (yr))
I 4
I
I
I
I 6
I
I 8
Fig. 2. Effects of varying parameters for a G/x= 10-6 string: simple gravitational decay as in fig. 1 (full line); superconducting decay Fo--0.1, 10 (dashed lines). Also shown are the increased low frequency power for simulations with small (c~=5× 10 -3) loops (dotdashed line) and high frequency power contributed by loops with and Pro~ m l decay spectra.
1 for reference (full line). In the model of explosive galaxy formation driven by superconducting strings [ 6 ] it is suggested that 10- 2 < Fo < 102. Spectra are shown for Fo = 0.1, 10, other parameters held fixed as above (dashed lines). Clearly for F0 ~ 1, large variations in the spectrum at 2GW ~ 10 kpc occur with varying levels of initial current. Also recent computations [ 5 ] suggest that the loops initially generated may be smaller, more convoluted and more numerous than in the standard picture. Taking rough values ( a ~ 5 × 10- 4 ]~ ~ 20, X~ 1000 [ 8 ] ) we compare the resultant spectrum for G/z= 10-6 (dot-dashed line) with that obtained from our fiducial values. Smaller initial loop size and larger total energy in strings enhance the emission at Pew ~ 104 yr substantially, in fact, the simulations suggest a range of initial sizes extending to even smaller scales and, if these are energetically important, the effects will be more pronounced. Finally, the effect of higher loop harmonics can be substantial, as shown by comparing curves computed for Pm~ m -2 and Pm ~ m - 1 (dotted lines). Note that for spectra shallower than ~ m -~/5 the 480
rise from loops surviving into the matter era spills substantial flux into higher frequency ranges. Loop radiation spectra with power law indices as shallow as - 1 would need a high frequency cut-off to ensure a finite total decay rate. p1/2
3. O b s e r v a t i o n a l
limits
The most significant constraints on the gravity wave background apply at wave periods of 1-10 yr (from millisecond pulsar timing) and at < 10 ~° yr (from microwave background isotropy). The current limits from timing of the millisecond pulsar P S R 1 9 3 7 + 2 1 [9] correspond to gravity wave energy densities of p < 1.7 × 10 -34 g / c m 3 at frequencies 0.8 < f < 2.2 y r - ~ and p < 7 × 10- 36 g / c m 3 at frequencies 0.23 < f < 0.63 y r - 1. These limits are plotted in figs. 1 and 2 as g?ow<0.38 (T2.7)-4f2v and ~GW < 1 . 5 × 1 0 - 2 ( T 2 . 7 ) 4~,~v,respectively. Although the pulsar timing model affects the gravity wave limits for periods close to the observation
Volume 215, number 3
PHYSICS LETTERS B
span [10], for a noise-free pulsar the bounds strengthen rapidly with time (approximately as the fourth power of the observation period). Following this trend in fig. 1, the bounds should reach < 10-4,Q¥ in ~ 10 yr, allowing one to constrain even Gp ~ 10 -s, superconducting strings. This sanguine prediction is, however, affected by two considerations. First, the stability of terrestrial clocks is rapidly becoming the limiting factor in the timing accuracy [9], which would slow the rate of improvement. Second, the presently quoted bounds are, however, very conservative in that they assume all residual timing noise is due to gravity wave perturbations. The recent discovery of several other fast, quiet pulsars will allow a cross-correlation analysis which uses the timing data of a number of pulsars in concord [ 11 ]; this allows both efficient removal of the terrestrial clock and ephemeris errors limiting the present analysis and effective discrimination against noise other than that due to gravitational perturbations. Timing accuracies for these other millisecond pulsars are presently ~ 3 - 1 0 times lower than for PSR1937+21. Several years of timing at these present levels would allow improvements of roughly an order of magnitude upon the single pulsar limits for ~Gw; improved sensitivity or additional quiet objects would allow much more effective use of the multiple arrival time data. Bounds on microwave background anisotropy can place important limits on the string energy density at horizon scales. As an example, in ref. [ 12 ] it is found that measurements at angular scales 0H 1 ° provide limits
G~t< (3/8zcv)1/2 sin-1/2 (½0) × ( 1 +zE) - ~ / 4 / f l ( ( A T / T ( O ) ) 2 )
1/2
~ 6 × IO-61AT/T(O) [ -5 sin-1/2(0/0.5 ° )
(6)
for v = 10 2, r = 10, ZE= 1500. The background isotropy also places limits on the gravity wave spectrum directly [ 13 ]. However, the waves at large scales are contributed by loops currently decaying and the strain on a scale l~ is hs~lsHo(3g2cw)l/2/c, where g2Gw is proportional to Cto/lszs at large scales and redshift zs. Thus the largest ls within the beam dominate the fluctuations and observations of A T / T < 2 × I O -5 on scales ~ 1 0 ° give the rough limit ~2Gw(P)< 1 0 - 2 ( ( A T / T ) 2 _ 5 ( O ) ) ( 10 s yr/p)2927 shown in figs.
22 December 1988
1 and 2. This is, however, constraining only at the present horizon scale, where the equivalent limit on G/z is comparable to that of eq. (7) above. At intermediate frequencies an additional bound may be of interest in the future. Timing of the orbital decay of the binary pulsar PSR1913+16 [14,15] provides limits on the gravity wave spectrum at P ~ 10-104 yr ofOGw < ½hff2(SPb/Pb)2_lO, where the present uncertainty in the rate of orbit decay is ~ 1.0× 10-~0(&fb/Pb)_,0 yr -~ [16]. If this may be improved to the point where the varying Doppler shift from a transverse velocity ~ 107 c m / s dominates the uncertainty then Ocw ~ 10Oy would be detectable. So while the computed background is maximal in the frequency range of this measurement (especially if a << 1 loops dominate the population), substantial improvement is required before such limits on string spectra compete with the background isotropy constraints. If strings emit substantial gravity wave power at frequencies < 1/ri, then present pulsar observations are sufficient to rule out strings with G# as large as 10- 6 when gravitational radiation is the principal decay channel. Similarly, from microwave isotropy measurements at ~ 10 ° it is clear that Gp must be less than ~ 3 × l0 -6. The constraints on the gravity wave spectrum from pulsar timing are more sensitive to the details of the decay, but the prospect of rapid improvement of these timing bounds suggests that limits on any reasonable scenario are attainable at wave periods of ~ 10 yr. The principal conclusion of the present study is that for such waves the spectrum produced by more realistic models of decaying cosmic strings will not in general follow the self-similar, constant fraction of £2 scaling; loop shrinkage, the preponderance of small loops and losses in nongravitational channels each produce important modifications. The spectra emitted by the individual loops are of particular importance in evaluating the observational bounds. If high frequencies are strong, the background at measurable frequencies would be substantially larger, and would be dominated by the loops normally important at ~ 10 kpc scales. These loops were born at the close of the radiation era and are of principal interest as seeds of galaxy formation (for strings radiating mostly in low harmonics, the background at yr-~ frequencies is dominated by loops formed near the epoch of nucleosynthesis). On the 481
Volume 215, number 3
PHYSICS LETTERS B
o t h e r h a n d , i f strings r a d i a t e p r e d o m i n a n t l y in the f u n d a m e n t a l , the b a c k g r o u n d s will be s o m e w h a t h a r d e r to m e a s u r e , b u t w o u l d s h o w s t r u c t u r e w h i c h w o u l d give i m p o r t a n t clues to the string d y n a m i c s a n d decay modes.
Acknowledgement We t h a n k Franqois Bouchet, G e o r g Raffelt, J o e Silk a n d A l b e r t S t e b b i n s for helpful discussions. C o m p u t ing s u p p o r t was p r o v i d e d by N S F g r a n t A S T 8615816.
References [ 1 ] A. Vilenkin, Phys. Lett. B 107 ( 1981 ) 47. [2] C.J. Hogan and M.J. Rees, Nature 311 (1984) 109. [3] T. Vachaspati and A. Vilenkin, Phys. Rev. D 31 (1985) 3052. [ 4 ] A. Albrecht and N. Torok, Phys. Rev. Lett. 54 ( 1985 ) 1868.
482
22 December 1988
[5] D.P. Bennett and F. Bouchet, Phys. Rev. Lett. 60 (1988) 257. [6] J.P. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231. [ 7 ] D. Garfinkle and T. Vachas19ati, 19re19rint ( 1987 ). [ 8 ] F. Bouchet, private communication. [9] L.A. Rawley, J.H. Taylor, M.M. Davis and D.W. Allan, Science 238 (1987) 761. [ 10] R. Blandford, R. Narayan and R.W. Romani, J. Astro19hys. Astron. 5 (1984) 369. [ 11 ] R.W. Romani, in: Timing neutron stars, eds. H. ()gelman and E.P.J. van der Heuvel (Reidel, Dordrecht, 1988). [ 12 ] J. Traschen, N. Turok and R. Brandenberger, Phys. Rev. D 34 (1986) 919. [ 13 ] E.V. Linder, Astrophys. J. 326 ( 1988 ) 517. [ 14 ] M.J. Rees, in: Gravitational radiation, eds. N. Deruelle and T. Piran (North-Holland, Amsterdam, 1983) p. 297. [ 15 ] B. Bertotti, B.J. Carr and M.J. Rees, Mon. Not. R. Astron. Soc. 203 (1983) 945. [ 16 ] J.M. Weisberg and J.H. Taylor, in: Proc. Millisecond pulsar workshop, eds. S. Reynolds and D. Stinebring (Green Bank, 1984) 19. 317.
PHYSICS LETTERS B
22 December 1988
LOW FREQUENCY GRAVITY WAVE SPECTRA GENERATED BY COSMIC STRINGS Roger W. ROMANI Depart,wnt of Astronomy, University c~fCali/brnia, Berkeley, CA 94720, USA Received 2 June 1988
Gravity wave spectra generated by populations of cosmic string loops are computed, showing that loop evolution and radiation properties have important effects at observable wavelengths. The spectra are compared with limits derived from various observations; bounds from millisecond pulsar timing are becoming severe for several string scenarios, but implied constraints on string parameters depend sensitively on details of the loop population and dynamics.
1. Introduction
An important aspect of scenarios which seed galaxy formation and produce large scale structure from the interactions of primordial cosmic strings lies in their testable predictions for an attendant stochastic cosmological background of gravitational waves [ 1 ]. At present the most sensitive constraints on this spectrum come from pulsar timing (at wave periods 1-104 yr) and from cosmological isotropy (periods ~ 10 ~° yr). Previous work has described the qualitative features o f the gravity wave spectrum for a population of idealized non-evolving strings [2], and calculated the total energy deposited in gravity waves during the radiation era for more refined models [ 3 ]. However, because of structure imprinted on the spectrum during the transition from the radiation era to matter domination, somewhat more realistic models of string evolution may have substantially modified spectra in observationally accessible regions. In this letter, we compute the expected stochastic gravitational wave backgrounds resulting from populations of cosmic strings radiating and evolving according to several string modes, considering a range of parameters that should delineate the expected behaviour. We find that there are indeed substantial variations in the shape of the spectrum for periods > 1 yr, and since present observational bounds are quite close to the predicted fluxes, such modifications are important for comparison with specific string evolution scenarios. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )
2. Gravity waves from loop populations We consider a population of long-lived (i.e. nonself-intersecting) loops following the cosmologically prescribed scaling behaviour which have a volume rate o f formation dni =Zt7 4 dti,
(I)
where a closed loop formed at time t, has a characteristic radius ri=c~ti, a typical total length fir,, and a mass per unit l e n g t h / t = e/G. These loops can decay through a variety of radiation processes; however, insofar as loops of interest are invariably massive, substantial amounts of energy will be deposited in gravitational waves. In computing the stochastic gravitational wave background one must, in general, consider the power spectrum o f the radiation emitted by each loop trajectory, average over all trajectories, average over an initial spectrum of loop sizes, follow the decay of the loops as they radiate and compute the evolution of the radiation spectrum to the present epoch. In the present context we are particularly interested in the shape of the resultant spectrum at low ( < y r - ~) frequencies, where evolution o f the loops and the cosmological expansion will be most important. The values of geometrical parameters, such as c~, and indeed the presence of a scaling solution, can be established by numerical simulations [4,5 ] and we will borrow values suggested by these studies. The gravitational radiation spectrum of an individual loop and evolution o f the loop trajectories also depend on 477
Volume 215, number 3
PHYSICS LETTERS B
details of the particle physics. However, the qualitative behaviour of loop decay can be simply written for a number of theories and we consider several models to display a range of forms for the final spectra. Several types of string are commonly discussed. In the standard picture emission of gravitational radiation causes string decay with an energy loss rate d E / d t = YGwG~ 2, where numerical work has shown that 7Gw ~ 50 [ 3 ]. In addition, strings can be "superconducting" and, if they thread sufficient flux, electromagnetic radiation losses from the persistent currents can be substantial. The magnitude of this emission appears rather more sensitive to the details of the loop trajectories than that of the gravitational loss channel. In particular, if discontinuities such as cusps or kinks are dominant sites of radiation loss, then the electromagnetic emission will be a roughly constant additional loss factor. The resultant spectrum will simply be that of a collection of loops with an increased ?Gw, renormalized to account for the lower gravitational radiation efficiency. However, if the electromagnetic losses are dominated by the low order modes of the loop, then the emission rate will be determined by the threaded magnetic field which will increase as the loop shrinks. This leads to a varying total radiation rate d E / d t = 7 ~ w [ 1 + F o ( r J r ) 2] , Fo = ( Io/ Icrit ) 2 ( YEM/ YGw )e2p2 / G lA ,
(2)
where p, ?EM/PGW~ 1 [6]. The initial current Io thus grows as the loop shrinks until a critical value Icrit is reached and it becomes energetically favorable to lose the topological twist in the field defining the string; the loop catastrophically decays into energetic particles or "quenches". To compute the gravity wave spectra resulting from these loops, we refer quantities to the era of matterradiation equality. Redshifting the initial loop density ( 1 ) to an epoch of gravity wave emission at z', and using the string density and emissivity above, one obtains the gravity wave emission rate deeq/dt' = (zeq/z')47GW
G/u2(z'/zi)3dni,
(3)
where the energy density is referenced to equality; Zeqm4.2XlO4(ff2hloo)-ZT4.7, with a Hubble constant of 100hloo k m s - l M p c -1 and a microwave 478
22 December 1988
background temperature of 2.7T2. 7 K. For a given wave frequencyfeq and a given radiating mode m, the time of emission t' determines the size of the radiating loop as r' = m [ 2 (z'/zeq )f~q ] - 1. For a loop radiating and shrinking as above this size can be related to the time of the string formation through the decay law (2), since d r / d t = - ( l / f l ~ t ) d E / d t . The radiation rate (2) gives an emission time of t'=ti
1 + ~
\at,]
-o
1-F~/2tan-lFff tan-
1
1/2
r'
which implicitly determines t' (the purely gravitational decay corresponds to F0 = 0). Taking the derivatives of (4) with respect tOfeq and t' allows one to solve for dfeq/dt'. The ratio of the emitted flux (3) to this quantity gives the gravity wave spectrum at equality as a function of the time of loop formation ti. For each frequency this may be integrated over the allowed range of G where ti.min corresponds to radiation atfeq at birth and t~.... is either the time of quench or the present time, to (for strings still radiating). The resulting gravity wave spectrum may be expressed as a fraction of the microwave background energy density ey= 3g/32Gt2q. Since the gravity waves redshift with the photons, ~2cw/~2v= (d6/dJ) eofeq/~v,eqand the present day spectrum as a function of gravity wave period is given by redshifting the wavelengths to to. Numerical simulations of evolving populations of loops in ref. [4] suggest Z~ 1, a ~ 0 . 1 and r ~ 10. Adopting these values as illustrative, we follow the prescription above to compute gravity wave the spectrum of a population of evolving strings. Gravity wave emission rates have also been computed for certain exact loop trajectories [ 3 ], for which in higher modes scales roughly as P,n ~ m-4/3. The relevance of these estimates for more general loop trajectories is uncertain; kinks may increase the loop emissivity while cosmological expansion and radiation back-reaction may serve to decrease the power emitted at higher frequencies. For the total wave power we use YGw-- 50; a similar total emission rate has been found for loops with kinks [ 7 ]. In fig. 1 we display spectra computed from the values with G/2=10 -6, 10 -s and F o = 0 (purely gravitational decay), Fo= 1.0 (fairly strong electromagnetic decay, with initial current
Volume 215, number 3
PHYSICS LETTERS B
'
'
'
I
'
'
'
I
'
'
'
p
.-""""" -2
I
'
22 December 1988
'
'
I
'
'
'
I
'
'
"' " " -
-4/a
V
BR\
PSR
........
* *
c
."" ~
,Gu = 10 -n
-3
"\
-;/
-4
- .........................
-a
J
.
.
I
.
.
i
I
I
2
. .""
I
i
I
I
i
0
I
/2~
I
_8
10-8
I
2
z
i
I
4
I
I
i
I
6
I
8
Log(P (yr))
Fig. 1. Gravitational radiation backgrounds generated by cosmic strings undergoing gravitational decay (G,u= 10 -6, 10-s; full lines) and "superconducting" decay (Fo= l; G,u= 10-6, 10-8; dashed lines). With a Pm~m 4/3 mode spectrum for the loop emission, the G/~= l0 -6 curves are modified as shown (dotted lines). Also displayed are bounds from pulsar timing and l0 ° background isotropy. 3 × 10-3Icrit). To display the change in spectral slopes for loops reaching toq these solid line curves show spectra for loops radiating in the fundamental only, in addition for the G/Z= 10 -6 strings we show the hardening of the spectra resulting from a loop spectrum P,, ~ m -4/3 (dashed lines). The qualitative features of such spectra have been described in refs. [ 1,2 ]. At high frequencies the most important strings are born and decay entirely in the radiation dominated epoch, thus channeling a fixed fraction of their energy density into gravity waves and giving a flat ~ p o spectrum. For 7GwG/Z<< 1, Fo>~ 1 and m = 1, eq. (2) may be approximated as t' ~ { 1 [2(z'/Zeq)~q]-3}tio'fl/3Fo~GwG/z. So the loop decay accelerates at t'*~tic~fl/3FoTGwG/z and the dominant contribution to the spectrum at f~q comes from loops formed at ~ 4o~2)c eq 2 t e q -/ I- , , . With these estimates, we can approximate the derivative d f ~ q / d t and the integral f (decq/dt') (dt'/dfcq ) dti, yielding
-Qe~w~t@a-zr(o/-fl)3/2(G/z/TGW)l/2(3Fo)-3/2g2v for feq >>
(fl/3FoaT~wG/z)l/2/2teq.
(5)
When Fo < 1,
(3Fo) - 3/2 is replaced by unity and eq. (5) agrees with the results of ref. [3]. At very low frequencies, the spectra follow £2GWOC(G/Z)2(p/to)-1 and the dominant loops were born in the matter era. However, at intermediate frequencies, the important strings are those born in the radiation era and surviving into the period of matter domination; accordingly the spectra in this region depend on the evolution and radiation properties o f the loops and simple estimates based on the loop lifetime and the total loop energy density may not be sufficiently accurate. Since 102 yr~> teqZeq/G/z , the waves in this region can be of interest to pulsar timing experiments. As the details of the string evolution are rather poorly known, constraints in this spectral region should properly be compared to specific scenarios. Conversely, if waves in this region should be detected, they might then provide some means of discriminating between various modes of loop decay. The importance of the uncertainties in the basic parameters is depicted in fig. 2, where we reproduce the purely gravitational decay, G/z= 10 .6 curve of fig. 479
Volume 215, number 3
PHYSICS LETTERS B
-
22 December 1988
,."" .....
.,'"
PSR
-
""-
a=SxlO-a
P m ~ m -2 ~
"'-,
\CBR
\
-1 __"
*PSR),". , ' .... ;
....
,
..............
-2 v O -1
t
-3
u
I
t
Fo=O.I
-4
-15 -2
I
I
,
I o
,
,
,
I 2
,
,
,
Log(P (yr))
I 4
I
I
I
I 6
I
I 8
Fig. 2. Effects of varying parameters for a G/x= 10-6 string: simple gravitational decay as in fig. 1 (full line); superconducting decay Fo--0.1, 10 (dashed lines). Also shown are the increased low frequency power for simulations with small (c~=5× 10 -3) loops (dotdashed line) and high frequency power contributed by loops with and Pro~ m l decay spectra.
1 for reference (full line). In the model of explosive galaxy formation driven by superconducting strings [ 6 ] it is suggested that 10- 2 < Fo < 102. Spectra are shown for Fo = 0.1, 10, other parameters held fixed as above (dashed lines). Clearly for F0 ~ 1, large variations in the spectrum at 2GW ~ 10 kpc occur with varying levels of initial current. Also recent computations [ 5 ] suggest that the loops initially generated may be smaller, more convoluted and more numerous than in the standard picture. Taking rough values ( a ~ 5 × 10- 4 ]~ ~ 20, X~ 1000 [ 8 ] ) we compare the resultant spectrum for G/z= 10-6 (dot-dashed line) with that obtained from our fiducial values. Smaller initial loop size and larger total energy in strings enhance the emission at Pew ~ 104 yr substantially, in fact, the simulations suggest a range of initial sizes extending to even smaller scales and, if these are energetically important, the effects will be more pronounced. Finally, the effect of higher loop harmonics can be substantial, as shown by comparing curves computed for Pm~ m -2 and Pm ~ m - 1 (dotted lines). Note that for spectra shallower than ~ m -~/5 the 480
rise from loops surviving into the matter era spills substantial flux into higher frequency ranges. Loop radiation spectra with power law indices as shallow as - 1 would need a high frequency cut-off to ensure a finite total decay rate. p1/2
3. O b s e r v a t i o n a l
limits
The most significant constraints on the gravity wave background apply at wave periods of 1-10 yr (from millisecond pulsar timing) and at < 10 ~° yr (from microwave background isotropy). The current limits from timing of the millisecond pulsar P S R 1 9 3 7 + 2 1 [9] correspond to gravity wave energy densities of p < 1.7 × 10 -34 g / c m 3 at frequencies 0.8 < f < 2.2 y r - ~ and p < 7 × 10- 36 g / c m 3 at frequencies 0.23 < f < 0.63 y r - 1. These limits are plotted in figs. 1 and 2 as g?ow<0.38 (T2.7)-4f2v and ~GW < 1 . 5 × 1 0 - 2 ( T 2 . 7 ) 4~,~v,respectively. Although the pulsar timing model affects the gravity wave limits for periods close to the observation
Volume 215, number 3
PHYSICS LETTERS B
span [10], for a noise-free pulsar the bounds strengthen rapidly with time (approximately as the fourth power of the observation period). Following this trend in fig. 1, the bounds should reach < 10-4,Q¥ in ~ 10 yr, allowing one to constrain even Gp ~ 10 -s, superconducting strings. This sanguine prediction is, however, affected by two considerations. First, the stability of terrestrial clocks is rapidly becoming the limiting factor in the timing accuracy [9], which would slow the rate of improvement. Second, the presently quoted bounds are, however, very conservative in that they assume all residual timing noise is due to gravity wave perturbations. The recent discovery of several other fast, quiet pulsars will allow a cross-correlation analysis which uses the timing data of a number of pulsars in concord [ 11 ]; this allows both efficient removal of the terrestrial clock and ephemeris errors limiting the present analysis and effective discrimination against noise other than that due to gravitational perturbations. Timing accuracies for these other millisecond pulsars are presently ~ 3 - 1 0 times lower than for PSR1937+21. Several years of timing at these present levels would allow improvements of roughly an order of magnitude upon the single pulsar limits for ~Gw; improved sensitivity or additional quiet objects would allow much more effective use of the multiple arrival time data. Bounds on microwave background anisotropy can place important limits on the string energy density at horizon scales. As an example, in ref. [ 12 ] it is found that measurements at angular scales 0H 1 ° provide limits
G~t< (3/8zcv)1/2 sin-1/2 (½0) × ( 1 +zE) - ~ / 4 / f l ( ( A T / T ( O ) ) 2 )
1/2
~ 6 × IO-61AT/T(O) [ -5 sin-1/2(0/0.5 ° )
(6)
for v = 10 2, r = 10, ZE= 1500. The background isotropy also places limits on the gravity wave spectrum directly [ 13 ]. However, the waves at large scales are contributed by loops currently decaying and the strain on a scale l~ is hs~lsHo(3g2cw)l/2/c, where g2Gw is proportional to Cto/lszs at large scales and redshift zs. Thus the largest ls within the beam dominate the fluctuations and observations of A T / T < 2 × I O -5 on scales ~ 1 0 ° give the rough limit ~2Gw(P)< 1 0 - 2 ( ( A T / T ) 2 _ 5 ( O ) ) ( 10 s yr/p)2927 shown in figs.
22 December 1988
1 and 2. This is, however, constraining only at the present horizon scale, where the equivalent limit on G/z is comparable to that of eq. (7) above. At intermediate frequencies an additional bound may be of interest in the future. Timing of the orbital decay of the binary pulsar PSR1913+16 [14,15] provides limits on the gravity wave spectrum at P ~ 10-104 yr ofOGw < ½hff2(SPb/Pb)2_lO, where the present uncertainty in the rate of orbit decay is ~ 1.0× 10-~0(&fb/Pb)_,0 yr -~ [16]. If this may be improved to the point where the varying Doppler shift from a transverse velocity ~ 107 c m / s dominates the uncertainty then Ocw ~ 10Oy would be detectable. So while the computed background is maximal in the frequency range of this measurement (especially if a << 1 loops dominate the population), substantial improvement is required before such limits on string spectra compete with the background isotropy constraints. If strings emit substantial gravity wave power at frequencies < 1/ri, then present pulsar observations are sufficient to rule out strings with G# as large as 10- 6 when gravitational radiation is the principal decay channel. Similarly, from microwave isotropy measurements at ~ 10 ° it is clear that Gp must be less than ~ 3 × l0 -6. The constraints on the gravity wave spectrum from pulsar timing are more sensitive to the details of the decay, but the prospect of rapid improvement of these timing bounds suggests that limits on any reasonable scenario are attainable at wave periods of ~ 10 yr. The principal conclusion of the present study is that for such waves the spectrum produced by more realistic models of decaying cosmic strings will not in general follow the self-similar, constant fraction of £2 scaling; loop shrinkage, the preponderance of small loops and losses in nongravitational channels each produce important modifications. The spectra emitted by the individual loops are of particular importance in evaluating the observational bounds. If high frequencies are strong, the background at measurable frequencies would be substantially larger, and would be dominated by the loops normally important at ~ 10 kpc scales. These loops were born at the close of the radiation era and are of principal interest as seeds of galaxy formation (for strings radiating mostly in low harmonics, the background at yr-~ frequencies is dominated by loops formed near the epoch of nucleosynthesis). On the 481
Volume 215, number 3
PHYSICS LETTERS B
o t h e r h a n d , i f strings r a d i a t e p r e d o m i n a n t l y in the f u n d a m e n t a l , the b a c k g r o u n d s will be s o m e w h a t h a r d e r to m e a s u r e , b u t w o u l d s h o w s t r u c t u r e w h i c h w o u l d give i m p o r t a n t clues to the string d y n a m i c s a n d decay modes.
Acknowledgement We t h a n k Franqois Bouchet, G e o r g Raffelt, J o e Silk a n d A l b e r t S t e b b i n s for helpful discussions. C o m p u t ing s u p p o r t was p r o v i d e d by N S F g r a n t A S T 8615816.
References [ 1 ] A. Vilenkin, Phys. Lett. B 107 ( 1981 ) 47. [2] C.J. Hogan and M.J. Rees, Nature 311 (1984) 109. [3] T. Vachaspati and A. Vilenkin, Phys. Rev. D 31 (1985) 3052. [ 4 ] A. Albrecht and N. Torok, Phys. Rev. Lett. 54 ( 1985 ) 1868.
482
22 December 1988
[5] D.P. Bennett and F. Bouchet, Phys. Rev. Lett. 60 (1988) 257. [6] J.P. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231. [ 7 ] D. Garfinkle and T. Vachas19ati, 19re19rint ( 1987 ). [ 8 ] F. Bouchet, private communication. [9] L.A. Rawley, J.H. Taylor, M.M. Davis and D.W. Allan, Science 238 (1987) 761. [ 10] R. Blandford, R. Narayan and R.W. Romani, J. Astro19hys. Astron. 5 (1984) 369. [ 11 ] R.W. Romani, in: Timing neutron stars, eds. H. ()gelman and E.P.J. van der Heuvel (Reidel, Dordrecht, 1988). [ 12 ] J. Traschen, N. Turok and R. Brandenberger, Phys. Rev. D 34 (1986) 919. [ 13 ] E.V. Linder, Astrophys. J. 326 ( 1988 ) 517. [ 14 ] M.J. Rees, in: Gravitational radiation, eds. N. Deruelle and T. Piran (North-Holland, Amsterdam, 1983) p. 297. [ 15 ] B. Bertotti, B.J. Carr and M.J. Rees, Mon. Not. R. Astron. Soc. 203 (1983) 945. [ 16 ] J.M. Weisberg and J.H. Taylor, in: Proc. Millisecond pulsar workshop, eds. S. Reynolds and D. Stinebring (Green Bank, 1984) 19. 317.